Lutris should have hash checks for file verification. There are big lists of file hashes from preservation groups like No-Intro and Redump.
When you install a game like Pokemon FireRed, Lutris should run a hash check on your selected file to verify it is correct. Side note: There is an American and EU version of FireRed but Lutris has only one generic option.
This is less-possible and less-necessary with PC games that have file integrity verification from Steam/GOG and/or multiple containers. Also, some people prefer to keep their ROMs zipped and dynamically extracted when played. In this case, Lutris could unzip the content first and then run a hash check.
SHA256 is the usual standard, but it might be more future-proof to use IPFS multihashes (in which case the multihashes would have to be derived from any reference files.)