Chloe asks a good question, Around 1900 both scenarios would have been common: some stayed put, some moved.
There are many reasons why our very mobile ancestors moved within the Russian Empire. These include (and this is not an exhaustive list):
1. Marriage: Shtetl level data for Salakas marriages - about 38 km down the road from Utena - for the period 1886 to 1910 shows that while most men married locally in the district or nearby districts, some married far more distantly. Recall there were customary reasons why often men moved to their wives' home towns. A summary of the data is at http://zarasai.blogspot.com/2009/08/more-on-marriage.html.
2. Conscription: Veterans often retired far away from their home towns.
3. Business success: First and second class merchants could live in the major cities: so one can perhaps find relatives born in St Petersburg and Moscow.
4. Opportunity: within the Russian Empire there were projects such as Shchedrin for farming which attracted Jewish settlers from some distance.
5, Education: whether secular or religious. At various times it was possible to get a University education.
Most of my Utianer relatives had emigrated by 1900 : South Africa, the US and UK were the typical destinations. There was an Utianer society in Johannesburg. It may be easier to find related Fishers abroad rather than elsewhere in Russia.
In any case, Chloe may want to first consider the odds of evidencing any link to Utian Fishers with any random Fisher she finds in Russia or elsewhere. A unified search for Fishers across the Jewishgen databases has more than 87,000 hits - while Fisher + Utena is only 78. "Fisher" may be too common to make fishing for them in the wider sea a profitable exercise.
The problem with searching for "Fisher" + Utena on most other databases is that the town name used in the record may not have been regularised for indexing, may look quite different and may therefore not be easily picked up - I have found documents that have Ootsyany, Utsiany, Utian, Utyan, and more - and random transcription makes this even more problematic.
An approach that combines documents and DNA might be most effective, if there are enough known Fisher cousins to test today to create a good benchmark for the DNA side.
SHADUR, SADUR, SHADER, SADER, CHADOUR, SADOUR, SHADOUR, SZADUR from Salakas, Lithuania
MINDEL, MINDELL from Utena and Vyzuonos, Lithuania
FELLER from Pabrade, Lithuania