ATHENS — War is never limited to where the fighting is. As trains and buses bring people fleeing Ukraine to the safety of border towns, mainly in Poland, they carry not just Ukrainians leaving a homeland under attack but large numbers of citizens of other countries who had made Ukraine their home and whose lives have also been upended by the violence.

Not everybody will make it to safety unscathed, and tragically some will not make it at all. On Monday an Israeli citizen was shot and killed at a roadblock by Ukrainian troops who reportedly misidentified him as a Chechen militant. The 42-year-old father of two, Roman Brodsky, was a deejay living with his family in Ukraine and had been traveling in a convoy of vehicles 60 miles south outside of Kiev heading for the Moldovan border; his parents told an Israeli TV station that despite the Ukrainians’ mistake they blame President Putin for their son’s death.

In a statement Monday evening, the Israeli prime minister, Naftali Bennett, said: “In the name of all Israeli citizens, I want to send my condolences to Roman’s wife, children and family in Israel and in Ukraine. We are doing everything to help Israelis return home.”

According to a statement from the Israeli foreign minister, Yair Lapid, 4,000 Israeli citizens have left Ukraine but thousands more remain. As of Tuesday morning, with Russian forces inching closer to the heart of Kiev, the state of Israel had positioned representatives at border crossings with Poland, Hungary, and Romania.

The U.N. refugee agency said Tuesday that some 660,000 refugees had already fled into neighboring countries from Ukraine.

“This figure has been rising exponentially, hour after hour, literally, since Thursday,” the agency’s chief, Filippo Grandi, told the United Nations Security Council. The Associated Press reported that Mr. Grandi said he has “rarely seen such an incredibly fast-rising exodus of people — the largest, surely, within Europe, since the Balkan wars.” The U.N. believes up to 4 million refugees could leave Ukraine if the war deteriorates further.

In Przemysl, a Polish town near the border that is the first stopping point for many refugees, the AP reported that there is a visibly large number of Africans and people from Middle Eastern countries.

Ahmed Ibrahim, a 23-year-old Egyptian, arrived carrying his cat in a carrier late Friday, feeling stunned and sick after days of travel. He said he had been studying medicine in Ukraine for five years and had only one year left. He had no idea what his future holds, not even what his next steps are. “What should I do?” he said.

In Przemysl, a Polish town near the border that is the first stopping point for many refugees, there is a visibly large number of Africans and people from Middle Eastern countries.

Ahmed Ibrahim, a 23-year-old Egyptian, arrived carrying his cat in a carrier late Friday, feeling stunned and sick after days of travel. He said he had been studying medicine in Ukraine for five years and had only one year left. He had no idea what his future holds, not even what his next steps are. “What should I do?” he said.