The cross-progression on Nintendo Switch is about to coming on Dead by Daylight, it is in fact undergoing internal testing as you can read from the following strings of code which show that it is activated in the internal build (thanks to N3emo):

The cross-progression was announced last year with an official release date as well, which was September 2020.

This new feature allows you to synchronize your game progression and your inventory (DLCs, purchases, skins) on several platforms.

At the end of 2020 it was actually made available but only between Steam and Google Stadia. And soon it will also arrive on the Nintendo console.

There are currently no plans for cross-progression between Microsoft and Sony consoles.