Make something happen when a parts position changes?

You used to be able to use the Changed event for this, but some time ago ROBLOX disabled it for properties that update too frequently. As in, they caused server lag when the even was connected. Part Positions were one of the properties cut.

You'll have to listen to some other event, for example the Running event of the Humanoid, and calculate the change in Position once it fires.

If you want to have the event fire when some arbitrary part moves, you're out of luck as far as I know, unless you can figure out some other event to listen to.

To add to Arbitrator

--If you want it to be for your character, you can use this event.

repeat wait() until game.Players.LocalPlayer and game.Players.LocalPlayer.Character
local p = game.Players.LocalPlayer
local c = p.Character
local h = c.Humanoid
local stopped = false 

h.Running:connect(function() -- The problem with this is that it also fires when you stop, but I fixed that.
    if not stopped then
        h.WalkSpeed = h.WalkSpeed + .5
        print("add")
        stopped = true
    else
        stopped = false
    end
end)