Diastereomers, enantiomers, and meso compounds of molecules with lots of chiral centres
Could we generalise that flipping an even number of chiral centres always gives either an enantiomer or the same molecule if it's a meso compound, and flipping an odd number of chiral centres always gives a diastereomer?


No. The generalization is slightly different: Comparing chiral centers in two stereoisomers with more than two chiral centres, if


all chiral centres are the same, it is the same molecule.
some but not all are switched, it is a diastereomer.
all are switched, it is an enantiomer (or meso compound).


Open-chain carbohydrates are a good example (compare D-glucose, L-glucose, D-galactose, D-talose etc.). For this class of molecules, there is one more term, epimer (when exactly one chiral centre is switched).

This generalization excludes the more exotic cases of stereoisomers, see e.g. Why are allenes chiral? and Why atropisomers are called conformers?