How (generally) do modern schemes deal with an adversary with the capacity to either instantly, or over time, corrupt a majority of parties
There is a model, typically called proactive security (and sometimes also called a mobile adversary), that considers the case that parties can be corrupted and later uncorrupted. Time is divided into epochs and security holds as long as the appropriate threshold of parties is honest in each epoch. In the dishonest majority case, this means that unless all of the parties are corrupted in a single epoch, security is preserved. These protocols work by rerandomizing secret sharing and enforcing correctness.