I've had to do this once with SAS and it worked out just fine. I called ahead to let them know and to ask what to do. Granted I wasn't transferring to another flight on the same day with a different airline - just needed to make my connection into a longer stopover. It's possible I've done it a second time needing a flight change but took a different flight with SAS... I still called them.

Luggage-wise they were totally cool with it. They have usually asked me when I have a connection if I want my checked luggage checked through all the way or if I want to pick it up at the connection. If they don't ask, just ask them yourself. They've been pretty accommodating about this with me as I'm sometimes actually dropping my bags in CPH for Le Boyfriend to pick up and take home before personally continuing on elsewhere without it- I have fun with luggage...