Car sometimes dies when shifting gears

Based on your observation the brakes are involved, I think I would pinch off the hose going to the power brake booster, then see how the engine acts. Be aware you won't have power assist for the brakes and the pedal will be hard to push, but you'll know very quickly if the symptom is still there. If the symptom is gone, start by replacing the check valve in the booster. The additional clue is right after you stop the engine, you should get two or three power assisted pedal applications before it gets hard to push. If you get no power assisted pedal applications, the check valve is leaking or the booster is leaking. If the check valve doesn't help, replace the booster.

That page won't open for me but booster check valves have such an extremely low failure rate, I'd get one from a pick-your-own-parts salvage yard. The chance of getting a bad one is pretty remote. All of these check valves do the same thing. They allow air to be sucked out of the booster and they prevent it from going back in. What you have to look at is the size of the large nipple that plugs into the rubber grommet and the size and number of ports on it that feed other things. Chrysler has used one valve for all of their models of cars and trucks for many decades. It wouldn't surprise me if Ford did the same thing. There's no need to change anything on this part. What Ford did differently than everyone else is they used a "vacuum tree" on the firewall that provided all the ports for other stuff. Some of them weren't used and they put rubber caps on them. Many other manufacturers use the port on the valve to feed the cruise control servo. If the vehicle came without cruise control, they put a rubber cap on that extra nipple too.