Most of history isn't a great guide for what happens next because of technology changes. Up until "recently" wars were won by logistics, numbers and training. In the modern age, logistics requires a very complicated industrial system capable of producing, eg, nuclear weapons and satellites. So far, only the US has pulled this off and only the EU has come close to (maybe they have managed as well, I don't know).
Life may get more tense when a new challenger to the US emerges. Maybe China.
China and Russia do wish to expand, but they're not going to do it while the US is in power at least.
The closest we've got is "spread democracy to the world", which is less a "conquer the world" spirit, and more of a "make the world like our way of thinking" cultural expansion. We do it through movies, literature, debates, exchange programs and so forth.
And... erm... conquest. But only rarely (Iraq, South Korea). And whenever we do it by conquest, the US Population seems to get really pissed in general. Its why we had such a "hands off" approach in Libya and Syria so far... and why we're trying to get Iraq out of our hair. (Iraq seems to be a "we broke it we bought it" situation at this point)
What evidence is there for China wanting to expand? It seems like they have made claims to disputed territories based on historical claims. They have a claim on Taiwan and and some disputed islands, all which had been under Chinese control in sometime in the past. That's not imperialism or colonialism. Look at Japan's behavior early 20th century. They took Korea, large parts of China and south pacific islands, things they never had a claim to and that is imperialism. Even Tibet had been under Chinese rule throughout centuries of history.
It seems like to me that China is only interested in achieving the same kind of cultural dominance and influence they've had through history. I've never read anything about China wanting to expand territory outside of traditional China.
Just because China once had some claim to certain seas doesn't mean they have any right whatsoever to them now.
Historically, yes, Crimea and Ukraine were all under strong Russian influence. But expanding to its historical size is an expansion from the current norm nonetheless.
Same with China. Do remember that Japan and Korea were historically claimed by China. Its unlikely that the US will let China expand to its historical size... and gobble up Japan and Korea while its at it.
I guess that's one way to put it. I could think of less favourable ways.