They should have to use their turn to stand or move.
Shove would be even more powerfull.
While I love the idea of being able to shove especially with such a verticality (to push but also to prone), the mechanic is absolutely overpowered.
You only have to look how much the ennemy is using it to understand that (to push you, but also to disengage as a bonus action, to wake up their sleeping allies as a bonus action, ...).
So +1 for shove to push + shove to prone as 2 different mechanics.
No to stay proned until next turn if you're pushed.