Blower motor

First of all, I didn't see a 3.8l engine in a caravan but did in a grand caravan.
The front control module controls the action of both the front and rear blower relays, so if the rear blower relay is behaving in the same manner as the front, then the problem is with that front control module, that is of course unless that delay behavior is normal. If it just happens once, it may be normal but if it just continues to repeat and start over and over, then perhaps there is something wrong with that front module.

If the ground has voltage on it, that means the ground isn't hooked up. If the ground flow through something else before it reaches ground, test again in and out of that item.
Give me a moment to get that wiring diagram.
I assume this is manual ac cause the auto ac blower motor wires are dark blue and dark blue/yellow, so here it is.
The black wire is the live wire and the dark green is the ground but it has to flow through that resistor first before heading off to ground and the ground is located at G200 behind the right side of dash, so check that ground for being loose or unhooked.

You are welcome.