Sorry, my Robert-Ned foster question was a bit vague. What I meant was, had lord Jon Arryn died after the 2 boys arrived at the Eyrie, would they be sent home or would they stay (be fostered) under Elbert Arryn or his Regent (should he still be in his minority)? Hope this clarifies TY :)

Oh, overwhelmingly likely the latter, for two reasons. 

First, as Jon Arryn’s heir, Elbert would be under strong pressure to uphold actions and decisions taken before his accession – if for no other reason that Jon Arryn’s decision to name him as heir would be called into question if Elbert started cancelling his decrees willy-nilly.

Second, there’s no reason for Elbert to send them back home. Having Robert and Ned as his foster kids is a huge honor that gives the Vale strong tires to the North and the Stormlands. Sending them back would be something of an insult to House Stark and House Baratheon, while gaining the Vale absolutely nothing.