WARSAW, Poland - Inna, 51, spotted the sign as she left the refugee center on the edge of Warsaw to go for a cigarette: “Help Ukraine! Give testimony!” it read.

“Help us punish the criminals!”

At first, she was not sure whether it was relevant to share what happened when her 26-year-old son left their home in the Kyiv suburb of Irpin in search of water. “There were others that suffered more,” she explained. “Nobody was killed except for the dog.”

But, with the idea that her testimony could be important, she sat down to recount her ordeal to a researcher with a 46-question form.

Three months since Russia began its assault on Ukraine, efforts to document war crimes committed during the conflict are hurtling ahead, both inside and outside the country.

As Kyiv investigates a mammoth 11,816 suspected incidents, prosecutors in neighboring Poland have gathered more than 1,000 testimonies from refugees like Inna who could act as witnesses.

France has deployed an on-the-ground forensic team with expertise in DNA and ballistics, and Lithuanian experts are scouring territory in eastern Ukraine. Meanwhile, the International Criminal Court, or ICC, last week sent in a 42-member team, the largest such contingent it has ever dispatched.

All together, it amounts to an unprecedented endeavor, experts say, and it’s happening in real time.

In no other conflict has there been such a concerted push to lay the groundwork for potential war-crimes trials from the start, said Philippe Sands, a law professor at University College London who was involved in the case against Gen. Augusto Pinochet, the Chilean dictator.

But the array of investigations - involving more than a dozen countries and a slew of international and human rights organizations - has raised concerns about duplication and overlap. That could result in “tension” between national and international bodies over jurisdiction, according to Sands.

Experts caution, too, that it could be years before any high-level decision-makers are held to account - if they ever are.

“The crucial question, the one that I think we ought to be focusing our attention on, is how do you get to the top table?” Sands said. It’s one thing to sentence a Russian soldier for killing a civilian, as a Ukrainian court did this past week. But establishing provable links between top officials and the horrors that have unfolded in places such as Mariupol and Bucha is difficult and time-consuming.

“This raises the specter of a situation where, years down the line, you’ve prosecuted a number of low-grade soldiers or conscripts for dreadful things,” Sands said. “But the people at the top table, who are truly responsible, got off scot-free.”

In an exhibition center housing more than 5,600 refugees on the outskirts of Warsaw, Inna paused to compose herself as she tearfully described her family’s ordeal in Irpin, while a volunteer from the Polish government’s Pilecki Institute for historical research took notes.

In the first days of the war, the power went out, followed by gas and then water. By March 8, the water situation was desperate and the family had run out of everything they’d stored. Inna’s eldest son left to seek help from a neighbor, but he was brought back by seven or eight Russian soldiers who accused him of spying.

When the family dog, Jimmy, went to greet them, a soldier shot the dog in the face, said Inna, whose last name was withheld for security reasons. “His lower jaw was destroyed,” she said.

The soldiers refused the family’s pleas to put the dog out of its misery, she said. Instead, they went inside and forced her sons and a friend staying with them to strip naked and lie down on the floor. “They were kept on the floor for around two or three hours,” she said. Eventually the soldiers left, after smashing phones and computers. The next day, the family risked the perilous journey out of Irpin, leaving behind Jimmy, whom they couldn’t bring themselves to kill.

“Do you remember what they were dressed in?” asks the volunteer, reading from the questionnaire. “Were they in uniform? Did you notice any special badges or patches.”

“Camouflage,” she answers. She can’t remember more. “Can it help anything?”

The Pilecki Institute’s Lemkin Center is gathering testimony both to serve as an oral history of the war’s atrocities and, if it might relate to a war crime, for referral to Poland’s public prosecutor.

The Polish prosecutor’s office said it has collected “very significant” testimonies from witnesses, alongside other evidence such as photographs and videos. “These activities are ongoing,” the office said, “they are extensive in nature, not a day goes by without us reaching new witnesses.”

Poland is one of 18 countries that have started their own criminal investigations into war crimes in Ukraine, according to Ukraine’s prosecutor general, Iryna Venediktova.

In the United States, where the State Department has asserted that war crimes have been committed by Russian troops in Ukraine, officials have said Washington could tap into its huge intelligence apparatus to assist investigations.

But with so many investigations underway, there is risk of organizations working at cross-purposes.

The U.N. special rapporteur on extrajudicial, summary or arbitrary executions, Morris Tidball-Binz, last week praised the international mobilization but urged countries and organizations to better coordinate.

“Without coordination of responsibilities and of efforts between various bodies, there is a considerable risk of overlap and duplication to the detriment of the effectiveness and efficiency of investigations,” Tidball-Binz said in a news release. “Proper coordination can also prevent the re-traumatisation of victims and witnesses arising from being interviewed multiple times by different investigators, and ensure that interviews fit into the overall investigative strategy.”

To reduce that risk, the European Union is adjusting the mandate of Eurojust - the bloc’s agency for judicial cooperation - to allow it to maintain a bank of shareable evidence, such as satellite images, DNA profiles, and audio and video recordings.

Poland, Lithuania and Ukraine have also signed up to be part of a joint investigative team alongside the ICC, meaning evidence gathered by prosecutors in any of those countries can be shared for national or international prosecutions. Estonia, Latvia, the Czech Republic and Slovakia are also in the process of signing up, according to Venediktova.

That partnership is key to building to what Venediktova describes as a “judicial front” in the war.

But others such as Germany - which is now home to 700,000 Ukrainian refugees and therefore many potential witnesses - are not coordinating directly, Venediktova said.

Always conscious of its own dark history, Germany has emerged as a hub for war-crimes trials in recent years. Using the principle of “universal jurisdiction,” which enables prosecution of crimes committed in other countries, Germany was the first, and so far only, nation to try an official from the regime of Syrian President Bashar Assad for crimes against humanity. A second lower-level official was convicted of serving as an accessory. In those cases, trials were possible because the perpetrators had ended up in Germany.

Germany has opened what it calls a “structural investigation” into war crimes in Ukraine, and in April, two former ministers, Sabine Leutheusser-Schnarrenberger and Gerhart Baum, lodged a criminal complaint against 33 Russian officials, urging Germany’s prosecutors to investigate them for war crimes.

“We are now urging the prosecutor to come forward very quickly, because the ICC is very slow,” said Baum, formerly minister of interior, noting that the ICC only just announced warrants for three Russian commanders involved in the war in Georgia 14 years ago.

Ukraine is not party to the statute that established the ICC, but its government has accepted the court’s jurisdiction over crimes committed on its territory and the country’s prosecutor general said her office will probably refer some cases to The Hague - whose mandate is to complement, rather than replace, national justice systems.

For Ukraine, the ICC’s involvement helps bolster the image of objectiveness, Venediktova said. It also can prosecute cases involving graver charges such as genocide and crimes against humanity - which cover large-scale systematic attacks, rather than individual acts. “What we see in Bucha and Irpin, it’s crimes against humanity,” she said. “That’s why for me their involvement is very important.”

Still, experts say whether any high-level officials end up in court could depend in large part on the political situation in Russia.

While the two former German ministers concede that the chances of Russian perpetrators ending up in Germany is unlikely, they said they hope international warrants might act as a deterrent on the battlefield.

Others disagree. “I don’t think that’s the logic the Russians operate on,” said Andreas Schüller, head of the International Crimes and Accountability program at the European Center for Constitutional and Human Rights. Since international sanctions have already restricted where Russian officials can go, international warrants, for now at least, would be symbolic to an extent.

Schüller said his organization, which worked extensively on documenting Syrian war crimes, is still sorting out how to be most useful on Ukraine. But it is still early, he said. What matters for the moment is less who is doing working on what, but that the documentation is happening.

“If you don’t start now, you will not have the opportunity in 10, 20, 30 years to act, if you don’t do your homework,” he said.

While the focus has been on Russian war crimes, rights groups are also working to document potential war crimes on the Ukrainian side, including the treatment of prisoners of war.

For Sands - whose 2016 book, “East West Street,” traces the intellectual origins of the Nuremberg war crimes trials after World War II back to the western Ukrainian city of Lviv - the key to getting more speedily to the “top table” of Russian officials revolves around prosecution of the lesser-known crime of aggression.

The Nuremberg Tribunal considered it to be the “supreme international crime” - the crime of waging the aggressive act itself. That, Sands argues, takes away the more difficult task of proving the intent of leading figures when it comes to atrocities on the battlefield.

Crimes of aggression are not under the jurisdiction of the ICC. So Sands has floated the idea of setting up an international tribunal to cover the crime. Since he wrote about it in a Financial Times column in February, the idea has taken off. On Thursday, the European Parliament voted for the E.U. to act to establish a tribunal.

“As things look right now, what are the chances of snaring one of the top people? No, it doesn’t look likely,” Sands said. But in 1942, people would have said the same thing, he added. “Of course, three years later, you know, Hermann Göring was in the dock at Nuremberg,” he said of the Nazi military leader sentenced to death in 1946 for war crimes, crimes against humanity and crimes of aggression.

The Washington Post’s Vanessa Guinan-Bank in Berlin contributed to this report.