Yeah, I just quickly looked and couldn't find anything on movement mods for attacks. (And why isn't there just a section on movement that can be quickly found in the index?)
My thoughts: Movement does not require a roll, unless there is a significant/immediate detriment to failing (Such as falling off a cliff) , or you ar tying to go further than normal. So, you don't make rolls for running across a field, unless tripping and falling can hurt you in some way, or you need to go a distance that is further than you can normally go. (I really have come to dislike the idea of putting scores of rolls in a game for things that really don't matter. Eventually, they will fail something incredibly inane and suffer horribly for it, at which point the game will truly begin to suck for them. You can rationalize all you want about "realism/versimilitude", but tripping while crossing a stream and losing your pulse rifle just before you reach the enemy is a terrible way for the "story" to go. IMO.)
As for movment granting DB, I would go with a simple +5 or +10 per movement rate over walking: Run +5/+10, Fast Run +10/+20, Sprint +15/+30, and Dash +20/+40. Which bonus depends upon if you are from the typical RM school of, "if it is for the PCs, if so, then it has to be the worst possible." I am more likely to go with the +10 per, because it is more friendly to the defender.