Hi. As I have said in other contexts, I am something of an expert in modern Russian and Soviet history. The answer to your question is no; Soviet citizens were not free to move from one region to another. However, all political rights (including the freedom to travel) to the extent that it was given, was always given preferentially to ethnic Russians.

The Soviet Union maintained an internal passport system which controlled movement. This was fairly easily adopted, because the right to travel was historically not extended to many in Russia. For example, in Imperial Russia, serfs were 'bonded' to the estate on which they worked. Though not slaves per se, serfs were nearly the functional equivalent. That structure was simply converted to accommodate industrial production as well as so called 'collective farms' during the period of Soviet Rule.

An internal passport system is a system, as the name suggests, required proper documentation and authorization to travel internally from region to region. This system was adopted in 1932, and reformed in 1953 when passports were made a legal requirement for all persons over the age of 16. The fact that internal passports were very difficult to obtain created an added institutional barrier preventing Soviet citizens from traveling internally within the USSR. Party leaders, on the other hand, or security service officers, easily obtained travel rights and many took advantage of them.