Probably not since they are not linked. 360 to One, ps3 to ps4

Highly doubt it, but it would be a cool feature.
However, if it did happen it would probably be like:

PS3 -> PS4
&
Xbox 360 -> Xbox One

However as new details arise about the Next-Gen ports we will know more information on this most likely.

You cannot transfer stats or anything like that from console to console as the coding is different and the way the game interacts with not only the ps4 system but the nextgen system.

BustySloth wrote You cannot transfer stats or anything like that from console to console as the coding is different and the way the game interacts with not only the ps4 system but the nextgen system.

I don't believe this is true, while yes the coding will be different for the Next-Gen versions of the games, considering it all pulls from a single database, it could very well be possible. But, most importantly difficult to pull off. It would work by syncing the progress of the different linked accounts of a single Rockstar account.

For example, if you have a Xbox 360 profile linked to a Rockstar account, as well as a Xbox One profile linked to the same account, there would be no reason for the coding to get in the way of the server performing that sync with the database, same goes for the PS3 and PS4 alternatives. Especially if the account is the same one, just on a different console.

Sorry for going in-depth and stuff, and this may not even be correct, but I don't see a reason it couldn't be done.

Yes you can, they said this at the Sony press conference.

Yes Exactly what Rustables said. during the conference Andrew House said ps3 and xbox 360 users can transfer their online data onto the ps4