We have a large collection of downloaded games from the store, mainly PSX gen (Crash Bandicoot, Spyro, FF7 etc.) and I was wondering if we can move them/re-download them onto the PS4?
You wouldn't be able to play any of them. PS4 doesn't support ps, ps2 or ps3 games. You can't re download them onto the console from the store for that reason. It is reasonable to assume that if Sony ever add support for, say, downloaded PS1 games then previously purchased titles will be available to you in the store.
You can't use a USB stick or similar to transfer them either as far as I'm aware, though since there's no point in doing so I haven't tried.
You can't move them over.
It currently looks like the PlayStation Now system will be the only way to play PS1, PS2, and PS3 games on the PS4.
Having said that, the PlayStation TV and PS Vita can play PSX titles from PSN.
Okay: I looked and looked as I am a ps3 owner I really want a ps4 but I just got the new game Call of Duty Advanced Warfare. I didn't want to buy a ps4 without knowing if I could redownload my game (s) to the ps4 the answer is yes you can but only if its from the psn store and if it says "Cross-buy" so if you have a ps3 and want to buy a ps4 yes you can download your previous digital copys from the psn to your new system.