The only thing that PSN will transfer is the account itself, probably the trophies and the PSN+ subscription, which you'll need to play online. Xbox One, nor the PS4 will have backwards compatibility, but Sony is working on a cloud service called Gaikai. With this service they plan to have games from PS1 to PS3 on their cloud, so you can play them on PS4.
Human Instrumentality Project
Sacrificing older life to create a new lifeform blessed with the fruit of life