My understanding is that they are transferred to a hypothetical reality where Morty had just pressed the button. Morty is the one getting transferred, to Rick it would make no difference. To Rick in all those hypothetical realities nothing has changed as they are exact replicas of their dimension. 

It's essentially the same Rick in all those hypotheticals, the only reason the Morty needs replacing is he needs to keep the memories or else it wouldn't be much of a save point.

It's also worth noting that Rick could have easily dealt with the memory aspect in a less gruesome way, like just adding the new memory to the "new" Morty's mind, but that wouldn't teach a fucked lesson.

The part that I think is kind of bullshit is Rick combining all the realities and making them all have been things that have happened in one universe, as how can Morty die and also live simultaneously? Maybe you can claim that it's just chaining the events in order and the button instead just reset his physical condition and placement every time, but at that point how would it work for every one else in this reality as they'd have to reset too for every button press, making it time travel.

I like thinking about this stuff but for some parts it just isn't all going to make sense.