
Think of PS3 / PS4 / PSV as a triangle, and saves can move in any direction (Vita saves are only applicable to BL2 and not BL:TPS).
The PlayStation family of systems can upload / download saves from any system to any other PlayStation system. The Xbox family of systems can only move saves from Xbox 360 to Xbox One and not the other way around. If you have multiple saves to transfer, upload one save, then download it in the Handsome Collection, then repeat the process for successive saves. IMPORTANT: Only one save may be uploaded at a time. Once the save has been uploaded, choose “Cross Save” or “Cloud Save” then “Download Save” from the main menu of the PS4 / Xbox One Handsome Collection game for which you’re downloading the save. From the main menu of the PS3 or Xbox 360 versions of BL2 or BL:TPS, choose the “Cross Save” (PlayStation consoles) or “Cloud Save” (Xbox consoles), then “Upload Save” function. Ensure all versions of the games are up to date with the latest patch (old-gen and new-gen) and you are connected to PlayStation Network or Xbox Live. Once you have that patch downloaded, follow these steps: To use a save from the previous generation versions of Borderlands 2 and Borderlands: The Pre-Sequel in Borderlands: The Handsome Collection, first you’ll need to download the patches enabling cross-save for Borderlands 2 and Borderlands: The Pre-Sequel, which should be available for download around the time of the Handsome Collection release.
If you've already played Borderlands 2 and/or Borderlands: The Pre-Sequel on Xbox 360 or PlayStation 3 (or PS Vita for Borderlands 2), you’ll be able to transfer your save within the same console family.