Archiving gmail

I have multiple devices - iPad, iPhone, iMac. I don't want a lot of old messages hogging memory on my phone and iPad but do want to keep them on my iMac.

When I archive emails on my iPad, for example, they disappear from all devices. Or are hidden someplace.

Two questions:

1) when I archive the message is it removed from the local memory on my iPad or iPhone?

2) how do I clean up the memory on my mobile device without losing messages forever on all my computers?

Running:

iMac OS 10.8.5, Apple Mail 6.6

iPad iOS 8.1.3, Apple Mail

iPhone iOS 8.1.3, Apple Mail