How do the ads get switched over during a site redesign or a new website with a new domain?

If it's the same website that's getting a re-design, setup tracking on all the new landing pages & forms during the week before the switch over takes place. If not, the only way to do it would be to pause the ads for a few days while you re-setup tracking.

Make sure you/your developers to make sure the same Google Tag Manager code and Google Analytics code get copied & pasted into the new design.

If it’s a new domain, then it’s considered a new website. It will need a new Google Tag Manager code and also a new Google Analytics “property”. Both sites will need to be up for about a week while you are switching over the tracking.