Hi, SGM,
The movement all starts at the beginning of the round for all parties involved. If an actor is making a "full move" and then melee, or half move and then missile or spell, though, the actions are resolved in phase 2. Phase 1: party with winning initiative goes first and takes care of all phase 1 actions; then the initiative loser (or initiative nonwinner for the sensitive!) knocks out all phase 1 actions. Then phase 2 goes the same way, with anything left over from phase 1 happening: the resolution of full-move melee and harge attacks, the resolution of missiles or magic after half moves; the completion of full moves.
A lot of this doesn't even need to be "public facing." The players tell you what they want to do, and you tell them when it all happens in the round.
But for actions by one party in one phase, I have it all happen at the same time. I don't break down the party by DEX--really, just in the order they're sitting around them table. Then I "give the kill" to one of them. Usually the one that will make the most people angry!