You simply take the next train. If you have savings fares which are bound to a specified connection, you need to obtain a written confirmation from the conductor on the delayed train (i.e. BEFORE the change) that the train is delayed, then you can use the next available train without any extra fees.

If you miss your train in Germany (Deutsche Bahn), your rights depend on the "high-fidelity" type of ticket you hold. If you have a Flexpreis ticket, you can simply hop on the next available train on the same route with no penalty. However, if you have a Sparpreis (Saver) or Super Sparpreis ticket, your ticket is "train-bound" (Zugbindung). If you miss the train due to your own delay, the ticket is technically void, and you must purchase a new one. Crucially, if you missed a connection because a previous DB train was delayed by 20 minutes or more, the "train-bound" restriction is automatically lifted. You can then take any subsequent train to your destination without needing a new ticket or a stamp from a conductor. For delays resulting in an arrival over 60 minutes late at your final destination, you are entitled to a 25% refund (50% for 120+ minutes), which can be claimed easily via the DB Navigator app or a physical "Passenger Rights" form.