My guess is there is sediment somewhere in the lines. The reason the flow decreases over time is the sediment is being dislodged when you drive as well as the fact that when the sediment dries it does not pack as tightly as it does when it becomes wet. Step one would be removing and cleaning all the screens on the faucets and running them wide open for a few minutes.
But there may also be screens on the lines at the water pump and maybe even obstructions where the lines are branched and connected. If you can't get to all of them, try attaching a hose to a faucet and back flush the system.