LH are generally good at keeping you informed of connections in your original flight if your flight is delayed, so you won't be leaving the plane without knowing what to expect. I have seen long haul flights hold off at the gate for late connections.
I had to do a 45mns Non-Schengen to Non-Schengen @
MUC once due to a late inbound flight. I was very worried at first, and in the end was so smooth I had time for a cheeky blitz pretzel and beer at the lounge.