Syncronised email client for multiple devices

I have been using Apple Mail on my MacBook Pro for years but have never been able to find the correct way to ensure that it is mirrored on my ipad or iphone.

By this I mean that when I reply a message it goes into sent folders (in real time or later over wifi) and if I move a file from the inbox into another folder that it goes across all three devices.

At the moment if I do something on my ipad, it doesnt show elsewhere. Someone has suggested Roundcube but I want to make sure that it is the right one before I move across.

I understand that a web based client is important to allow all this to happen. I have a Yahoo account and IMAP accounts all coming into Apple Mail at the moment.

Any help would be very much appreciated.