ESA does not really compete with NASA. In fact, they collaborate quite a lot. The ESA also collaborates with Roscosmos to a lesser extent through the 'launcher' arm of European governments (Arianespace) which collaborates with Russian launch vehicle providers quite extensively. They now launch 'Europeanized' Soyuz rockets from Kourou, for example. ESA and Roscosmos are also collaborating on a Mars orbiter/lander though this is the only scientific collaboration outside of the ISS I am aware of.
NASA isn't really falling behind in exploration and by many measures is doing better than just about everyone. The ESA restricts itself mostly to the ISS and LEO (with a few notable exceptions such as Rosetta) and Roscosmos does almost nothing scientifically speaking aside from experiments on the ISS.
Meanwhile, NASA is gearing up for manned exploration of deep space via the SLS, has begun the process of handing over astronaut and resupply flights to private companies (which can then use the tech they develop to go deeper into space - see SpaceX's mission statement) and regularly launches scientific probes into deep space, other planets and LEO. The corner also seems to be finally turned on chronic under investment in NASA by politicians who are now finally giving funds in excess of what NASA requests.
As for China, their plans are not well known to the western public and what little information we do know frequently changes. They are known to be working on a large booster for Lunar exploration but the progress of that booster is anyone's guess. They are committed to launching more space station modules of their own as well as more Taikonauts and have expressed desire to do manned lunar landings. Given the secrecy of their space program though, how much progress they are making is real versus pure propaganda is unknown to a large extent.
Russia, by contrast, has slashed the budget of Roscosmos quite drastically and has scaled back plans for Lunar and Martian exploration to the point where they probably won't happen at all now. Further, they've finally shut up about detaching their modules from the ISS and starting their own station with them. They clearly can no longer afford that and it never worked as a political bargaining chip in the first place because despite the fact that they are the current sole launcher of manned missions to the ISS, that's about it. Their entire space program is derelict outside of their ISS involvement so for prestige reasons, they have to continue supporting the ISS as-is. They are working on a new launch pad in the Russian Far East (Vostochny) but that's seriously over budget and schedule, has been plagued with corruption and really only serves as a prestige project since it provides almost no real advantage to space exploration for them. Additionally, they are said to be on the cusp of introducing their new Angara rocket (which is also decades overdue) but it's yet to be seen that they will truly follow through with that and also what benefit the new launcher brings them over their older ones. I guess it may allow them to retire the Proton, (which is a mess on many levels) but only after the 'heavy' variants of Anagara are introduced which again, are probably a decade or more in the future, if at all.
