Sort of. Playonlinux has better functionality for transporting games, but Lutris has better game compatibility.
What you can do, is move your game prefixes to a folder on your new PC, and then sign up for a Lutris account, and add the games to your Lutris account from the Lutris game library:
Then sign into your Lutris account with the Lutris app on your new PC, and the games should be listed now in Lutris, and you can right click them and go “add installed game”.
From there, you have to manually set up all the Lutris preferences for the game, which is a hassle if it’s a complex setup, but not too bad if it’s a simple one.
It’s usually not too bad setting everything up by looking at the installer on the game you’re setting up, and just copying everything it does manually using the Lutris gui. Here’s an example:
If you click your game, and then click the “view on Lutris.net” link on the right side of the window, that opens the game page in the web browser, and then you find the installer you want, and click the down arrow next to the install button, and select “view installer script” from the down menu.
It’d be nice if the Lutris team would make some kind of system to set up all the settings from the install script for you, but it’d be tricky, because probably you’d have to create separate import game scripts for each install script, and I can imagine it’s a lot of work upkeeping just the normal install scripts as they are.
Good luck with it
Probably a better solution in many cases:
Another route you could take, is to install the games as normal, but just copy the game install files rather than the whole wine prefix. You’d just need to select your game directory when the game installer is installing. Most game installers will CRC check, or just skip existing files. So the install would be quick in that case.
It should be possible with Overwatch and Starcraft to. Probably the Witcher 3 gog installer would do something similar.