Cranks but will not start after it has run for any amount of time

You're going about this the wrong way. Replacing random parts is by far the most expensive and least effective way to diagnose a problem. It would be much less expensive to have a mechanic do that for you. Instead, you have me, and together we should be able to solve this.

The place to start is by reading and recording the diagnostic fault codes. Chrysler made doing that much easier than any other manufacturer. Cycle the ignition switch from "off" to "run" three times within five seconds without cranking the engine. Leave it in "run", then watch the code numbers show up in the odometer display.

This doesn't work on newer models. If it doesn't work on yours, you'll need a scanner or an inexpensive code reader. You can go here:

https://www.2carpros.com/trouble_codes/obd2/p0100

to see the definitions, or I can interpret them for you.

The next problem is the relevant fault codes often do not set from just cranking the engine. Specifically I'm referring to codes for the crankshaft position sensor and the camshaft position sensor. Missing or lost signals from them need a little time to be detected, as in when a stalled engine is coasting to a stop. For that reason, don't assume a sensor is okay just because it hasn't set a fault code.

Additionally, one of the more common ways for these two sensors to fail is by becoming heat-sensitive. They'll work fine as long as natural air flow over the engine keeps them cool, but when a hot engine is stopped, heat migrates up to them, causing the temporary failure. They'll commonly work again after cooling down for about an hour. As this problem progresses, the sensor may fail while you're driving, especially in warm weather.

Let me know what you find with the fault codes.

Not running hot. It's normal under-hood heat from the engine that heats up all of its sensors, but it's just the cam and crank sensors that become heat-sensitive. We used to run into this all the time in tvs and vcrs. All we could do was cover them up with rags and a lamp, then wait for the intermittent problem to occur. Once something started to act up, we used a spray can of "Freeze Mist" to cool down very specific parts, one at a time, until the problem changed or cleared up. Unfortunately that doesn't always work on cars, especially when you have to drive it until the problem occurs, then run alongside it with the hood open, chasing it with that can of Freeze Mist".

Some sensors, like the older throttle position sensors, are strictly mechanical sensors that aren't affected by temperature. Some are magnetic with a coil of wire wrapped around a magnet. Older cam and crank sensors used to be of this design. They could be identified by having just two wires in their electrical connectors. Those could fail too from under-hood heat, but is was much less common. The very thin wire would expand and contract repeatedly with each engine warm-up cycle, which eventually led to a wire breaking off its terminal. Often those would work again as the wire and terminal cooled down, and that wire would just barely touch the terminal. Here again, natural air flow while driving usually kept them cool. It wasn't until the hot engine was stopped that the lack of air flow caused the heat-related problem to show up.

Cam and crank sensors with three wires are those that have delicate electronic circuitry inside them. Heat is always the deadly enemy of electronic components. Mass air flow sensors, used by every manufacturer except Chrysler, and MAP sensors, both have a lot of circuitry inside them, but they are both near constant air flow, so heat-related failures are less common. Even the heat from a hot, stopped engine rarely affects them because of where they're located away from the areas of high heat.

When you don't have any related diagnostic fault codes to indicate which circuit needs further diagnosis, you have to use a scanner to view the live data the computer is seeing. I have a Chrysler DRB3 for all of my older vehicles. It lists the crankshaft position sensor and the camshaft position sensor with a "No" or "Present". As soon as you start cranking the engine, they should immediately switch from "No" to "Present". When one of them remains on "No", that is the circuit that is not developing its signal. When either signal is missing, the Engine Computer will not turn on the automatic shutdown, (ASD) relay. That relay powers the injectors, ignition coils, alternator field, oxygen sensor heaters, and fuel pump or pump relay. All manufacturers use some variation of this system. If a fuel line gets ruptured in a crash, there would be a serious fire hazard from the raw gas being dumped on the ground. Instead, there can't be any fuel pressure in a broken fuel line. With no fuel pressure, the engine can't run. Once it stalls, there's no signal pulses from the cam and crank sensors. That tells the computer to turn the ASD relay off, thereby stopping the fuel pump.

Nothing is solved by "rebooting" the computer. This thought goes back to late '80s and early '90s when GM had a really huge failure rate with their Engine Computers. Almost every problem you could think of might be solved by replacing the computer. At the same time, Chrysler's computer failure rate was so low, they were always the last thing on the list of suspects.

Engine Computers start out with factory-installed software that learns the characteristics of every sensor. When a sensor is replaced, there are strategies for it to learn the new characteristics by comparing it to other known values and operating conditions. When a battery cable is disconnected in a misguided attempt to fix something, all of the sensors have to be relearned along with a real lot of fuel trim data. Short-term fuel trim numbers update continuously as you drive. When the computer sees it is always making the same adjustments, it moves those numbers to the "long-term fuel-trim" memory. Those LTFT numbers are updates from the starting values programmed in at the factory as default values. All that learned data is lost when the battery is disconnected.

The learned sensor personalities involve those that develop a signal voltage proportional to temperature, position, or flow. Examples are the coolant temperature sensor, intake air temperature sensor, battery temperature sensor, throttle position sensor, EGR valve position sensor, mass air flow sensor, and MAP sensor, (vacuum). The crankshaft position sensor and camshaft position sensor are not of these types. They develop square waves that switch from 0.0 volts to 5.0 volts, but it's not the exact voltages the computer cares about. These are used for timing, so it's the point at which one switches from "low", around 0.0 volts, to "high", around 5.0 volts, that's important. Those switching points tell the computer when to fire injectors and ignition coils. Those don't change when replacing one of those sensors. The timing is determined by notches, or windows, cut into the metal ring around the flywheel or torque converter, or from teeth on a disc on the camshaft and crankshaft. Those holes or the gaps between the teeth disturb the magnetic fields the sensors create, and it's that movement of the magnetic fields that induces a small voltage that the sensor's circuitry amplifies and sends to the computer.

With all those other sensors, no two are ever exactly alike. That's why computers are designed to learn the characteristics of new ones. With the cam and crank sensors, being alike never enters into the story. They all respond to the mechanical parts that trigger them, and those parts don't change when a sensor is replaced.