It is a long story. I am not completely sure which period you are actually interested in. Mensheviks (RSDRP) ceased to exist as a party in 1951, so I will address the earlier period, up to the 2nd World War. RSDRP in immigration was split in the left and right wing. Roughly speaking, the left (Dan, Gurevich, Schwarz, et. al) following "Martov Line", were hoping to find a common ground with the "moderate" bolsheviks (like Bukharin), they were also hoping for an evolution of the USSR into something more resembling socialism (as they understood it). They were also considering the economic developments in the USSR as the developments of the "state capitalism", which was supposed to, eventually, lead to true socialism. The right wing (Potresov, Gravi, Aronson, et al) , in contrast, were hoping for a popular uprising in the USSR (once things turn really bad - and they did, especially during collectivization). The right wing of RSDRP regarded Soviet system as a form of fascism and did not expect its evolution into a an actual socialism. By the end of the 1938 (after Moscow show processes, mass terror, etc.), and especially in 1939 (after Molotov-Ribbentrop pact) both wings realized that they were wrong. RSDRP in immigration understood very well that their underground organizations in the USSR were completely wiped out. Thus, by publishing in the "Socialist Messenger", Mensheviks were primarily talking to each other (and their numbers were tiny), it was a way to flash out their ideas and disagreements. Their only semi-realistic way to effect the events was through Socialist International (RSDRP was part of it). However, due to their internal split, Mensheviks could not accomplish even that. I am not sure what RSDRP position during the WWII was, if I had to guess, they were again split on how to support the Soviet Union in this situation (i.e. support 100% or support, but to condemn the Soviet totalitarian system at the same time).
Incidentally, after the WWII, some of the Mensheviks, e.g., Dallin and Nikolaevsky also known as Nicolaevsky, again became relevant. Among other things, they interviewed Soviet DPs who found themselves in the Western Europe, to record the plight of the ordinary Soviet citizens under Stalin's rule (D.Dallin, B.Nicolaevsky, "Forced Labor in Soviet Russia", YUP, 1947). Since Mensheviks were acutely interested in what was happening in the USSR, they turned out to be useful in the beginning of the Cold War. For instance, once the Russian program of Radio Liberty was established in 1950s, it would invite Mensheviks for their broadcasts (rather than, say, monarchists), I can explain why if this is interesting. The same Nicolaevsky (who immigrated to the US in 1940) was one of the first American "sovietologists" and is regarded by some as "the father of Kremlinology" (see "The Anti-Communist Manifestos: Four Books That Shaped the Cold War", by John V. Fleming, 2009).
There are several sources you may want to take a look at, for instance, "From the Other Shore: Russian Social Democracy After 1921",
by André Liebich, HUP, 1997.
Most of my sources are in Russian, so I am not sure if you will find them useful.
There was another party you may want to read about, the Socialist-Revolutionary Party, more precisely, its right wing. In exile, they continue to function through 1940s.