It may have been the Russians, but it was probably the Austrian Army they were dodging.. At this time, Poland did not exist as a country after the many partitions that began in 1794. Poland was split between Prussia, Austro-Hungary, and Russia.
I know that in the Austrian partition known as Galicia, in the late 1890s and early 1900s the Austrian Army was drafting people. Polish males did not want to serve in a foreign army so a lot of them fled the country.
I have spoken with descendants of Polish men who emigrated to New York state in the early 1900s. Sometimes whole villages of men emigrated to the USA. Some were not so lucky. One man told me that his Polish father was conscripted into the Austrian Army and was immediately transferred to Italy where he worked more like slave-labour in a dolomite mine for 4 years.
There is a very well-known Polish movie called 'CK Dezerterzy" ('His majesty's deserters') based on this period.