Say I don't use my tablet for a while. I deal with all of the notifications on my phone as they come in. I then turn on my tablet, and all of these notifications ping off in rapid succession.
Are you actually acting on the notification besides swiping it away? If you only swipe it away that action will not sync. If you read an email it should disappear (or never appear) when you look at the tablet.
Welcome to Android Central! This is pretty standard behavior. I think if you wait a while, at least some of those notifications will disappear as the apps sync with the cloud. But it won't always be immediate.