It almost sounds like you might have "air" getting built up, in the fuel system (or air is getting built up, in the fuel filters) then it starts to get "pushed out" while driving , but it takes a while for all that air volume to get thru the complete fuel system {couple miles of driving), and then finally, the fuel filters get mostly fill of fuel again (then the engine runs good, again) This is just a guess, but it would require some fuel sight glasses to prove this "air in fuel" type of problem.

Here's where to plumb a fuel pressure gauge:

(3406E fuel filter base_fuel pressure port 2.jpg)

3406E fuel filter base_fuel pressure port 2.jpg (35KB - 521 downloads)