Yes it works, there's a bunch of standards (XEPs) describing how a client/server should handle unreliable connections, push notifications, etc. However, not all servers respect these. If you're setting up your own, it's easy to enable though.
Also a problem is iOS clients. Developing on iOS requires lock-in into the Apple ecosystem and most XMPP folks are free-software people so nobody will bother to buy Apple hardware/software specifically to develop for their closed ecosystem (i.e. unpaid labor for a multi-billion corporation), although i'm sure a lot of folks would be happy to do just that if that was sponsored work (donations, grants).
On Android, Conversations (or its forks Quicksy/Snikket) is the best. ChatSecure on iOS has been buggy for years (every time i tried with someone with an iPhone), but nowadays Siskin appears to be a good iOS client, though i haven't had the hardware/system to try it myself.
Also a problem is iOS clients. Developing on iOS requires lock-in into the Apple ecosystem and most XMPP folks are free-software people so nobody will bother to buy Apple hardware/software specifically to develop for their closed ecosystem (i.e. unpaid labor for a multi-billion corporation), although i'm sure a lot of folks would be happy to do just that if that was sponsored work (donations, grants).
On Android, Conversations (or its forks Quicksy/Snikket) is the best. ChatSecure on iOS has been buggy for years (every time i tried with someone with an iPhone), but nowadays Siskin appears to be a good iOS client, though i haven't had the hardware/system to try it myself.