Mail Rules on Multiple Devices

I have a MacBook pro, an iPhone and an ipad. I use Rules in my Mail application on my MacBook pro to sort everything that comes in (many rules to many folders). I use a Hotmail account. I'm not sure what changed in the last few months, but now if I leave my mail application open on my mac at home, the mail goes to its proper folders, but then disappears from my iPhone.

Example: emails from mom, go to family folder. If I happen to see one come in while on my iPhone, I see it appear, and then quickly disappear. When I get home, I see the email on my mac, moved to the family folder (like it's supposed to).

This is an issue, because I often leave my mail app open at home, and then I cannot see most of my emails when i'm away from there.

Help!