There have been in history. Vietnam for example. But now that China has abandoned ideological communism more or less, and become wealthy what proxy battles are going on with them?
I don’t know of any active wars at the moment. I know the north Korea situation is a proxy issue, and the US and China are competing for African natural resources but again, not a war.
Is there competition for influence in Latin America between the two nations?
The US and China are hacking the hell out of each other’s computers. I guess that’s not really proxy fighting, though, but direct espionage and vandalism.
The US has had several successes in jockeying for influence in countries around China lately, but that’s diplomacy rather than combat.