There is an actual observable phenomenon sometimes when you switch brands. For a period of time, sometimes a couple of changes, consumption may increase. There is a theory that a different additive mix causes one chemistry on engine parts to shed and another to replace it. The term used to describe this is "seating". Again, it's the theoretical explanation for something that is seen once in awhile.
Another issue that arises is a more detergent oil replacing a less detergent oil, or an oil that causes a bit less seal swell replacing an oil that swells seals more. In both cases you may get some leakage, either for awhile or permanently. Either the gunk holding the oil dissolves, or the seals shrink.
These days both of these are uncommon but not unknown.
The biggest problem with changing brands around is that if you're doing oil analyses you can't get any sort of baseline. It's the changes from baseline that are meaningful in oil analyses, and you can't get a baseline if the additives are changing all the time.
So, generally changing brands is not a problem, nor is changing viscosities, if you use a good quality modern motor oil. Lots of people run cars for years to high mileages using whatever API rated motor oil in the weight they prefer that's on sale when they change it.