Soyuz is a spacecraft which still uses technologies from soviet era. The Dragon capsule is state of the art with its electronic upgrades. With Russia and US sharing the International Space Station along with few other countries, do the non-American astronauts and cosmonauts train on Dragon?
At time of this writing only one Crew Dragon has flown, so tense shall be future tense.
There are two ways to handover crew at the ISS. Direct, where a new crew launches and there are three crews on station for a week or so, and then the old crew departs. There is also indirect transfer where one of the crews departs and the new crew arrives a few days later.
There is a desire to always have 1 Russian and 1 American on station at any given moment in time. Thus in an indirect handoff (which seems to be the current option for the last bunch of handovers) one of the vehicles be it Soyuz (from 2011-2020 the only option) or Dragon or CST-100 is leaving. Which means if Soyuz only carries Russians and Dragon/CST-100 only Americans, then the station could be crewed without a mixed crew. (Europeans, Canadians, and Japanese etc figure out which one they fly on and can be ignored for the purposes of this conversation).
The Americans and Russians are trading seats on each vehicle going forward to keep to this standard. Thus Americans will still fly in Soyuz and thus require training. Russians will fly as part of the crew on a Dragon or CST-100 and thus require training.
A more interesting question is how much cross training is there on any particular crew? I.e. Does an American arriving on a Soyuz have to train on how to land in a CST-100? To what extent? And the converse?
After some successful test flights with two american astronauts there will more tests with more than two astronauts.
After all those tests were done, selling of seats to non american astronauts may follow. But training of non american astronauts may be delayed by some months up to years by Covid-19 disease.