Russians, or Prussians? I ask because the Six Weeks' War in 1866, which was largely fought between Prussia and Austria, saw much of its fighting in Austrian Bohemia, notably the Battle of Sadowa/Königgrätz.

Also I should point out that Bohemia and the Russian Empire did not share a common border in the late 19th century: Prussian Silesia was in between the two, and the Prussian, Austrian and Russian borders met at the so-called Three Emperors' Corner in Polish Silesia, not far from Krakow.

My answer may not help a lot, just wanted to point there weren't any open conflicts between the Austro-Hungarians (Bohemia was part of Austria-Hungary in the 19th Century) and the Russians. However, they shared frontiers at the Northern Sudetes, so the backstory may not be out of place at all. Some crime made in Russian territory, an offense to a Russian authority, and an accomplice from the Austro-Hungarian one could lead to an exile. Dunno, just theorizing.

There were conflicts with the Slavs folks during all the existence of the Empire, a dispute of power between the German-Austrian authority and all the minorities (Hungarians at first, Czechs, Slovenes, Italians, Croats, Serbs, and a couple more). An act of rebellion against the goverment could earn that exile too.

However, you're pointing out they arrived to the US as refugees. There were economic fluctuations during the late 19th Century, the most important being the panic of 1873, which even had a bigger impact in the States than in Europe, but still, Austria-Hungary was hit by the depression. Maybe your family lost it all in those years.

Thank you