In the case of the 16 bit systems, they just didn’t have the overall power to handle everything flawlessly. Sprites had to be downsized, and limited memory caused some frames and animations to go missing.
The PS1 lacked the RAM necessary for flawless ports of the newest fighters like Alpha 3 or Marvel vs Capcom, while handling simpler ones like SF2 great and even Alpha 1 pretty well.
The Saturn handled most 2D ports great, mostly due to the extra RAM.
Since then most ports have been spot on (most). However, changing over code from one system to another isn’t always easy or flawless. It takes time and care to get it right. If the code is ported sloppily. the game will suffer. Likewise, sometimes a company tries to “fix” things in a home release that players don’t like (see Capcom vs SNK 2 EO).
If you want to see a port done right, play Alpha Anthology for the PS2