Sorry if this is something that has been addressed or detailed somewhere else, I just couldn't find the sources.

I have noticed in new speedrun footage there are particular times that the runner jumps between screen transitions. I was just looking into more details on that Strat

Hi Theduff, great observation! It was only recently discovered that the Player can enter a 'glitched' state where each punch, kick, weapon strike and weapon throw (but not enemy throw) deals an additional 1-2 damage. You can perform this glitch in either River City Ransom or Downtown Nekketsu Monogatari by simply landing a high jump and it will remain active across screens until the Player next leaves the ground by jumping, hopping, or falling.

Yes, further to what will said, executing the jump is enough to enter the glitched state, and it doesn't matter where you do it. We do it going into screen transitions because jumps are slower than running, so we get the benefit of the glitched state without having to wait for the end of the jump animation!

There are a couple of other spots where we lead into a boss fight with the big jump, which serves two purposes. The glitched states increases your damage output, but the jumping punch attack also does more damage than a running punch attack, which is used to reduce the number of rounds needed for the Otis fight, for instance.

Thank you both so much for the detailed explanations!