There's a sub panel in each building feeding from a main panel?

I really don't think its your cables under ground. If you want to rule that out, just disconnect the wires as soon as it enters the building and go and turn the breaker on. There should be your answer.

OK Terminology. All wiring downstream of a breaker is a branch circuit. I think I understand your setup is all branch circuit but you are differentiating the underground feeder as a supply and the above ground wire in the buildings as branches. I would have set up a subpanel in each building or at least a seperate branch circuit for each not a single breaker serving more than one building (which is illegal here also) and then trouble shooting would be a little simpler.
You need to isolate parts of the circuit until you find the fault. I would ,
1) discon everything at the breaker to test the breaker.
2 then discon the individual buildings one at a time at the box where you enter to decide if the fault is underground or in the building.
3) When you determine if in say building #2 there is a fault ,then you start at the end of building #2 and systematically discon stuff there until you locate the fault(either a pinched wire in a wall between box 6&7 or a bare wire in a box touching another or something will appear.

You could use and OHM meter or meager! On the branch circuit you disconnect Make sure power is off and every thing is unplugged and switches off,connect the ohm meter to hot and neutral of the branch circuit, there should be no connection or infinity. And do the same with each wire to ground,there should be no connection to ground either. Do one branch circuit at a time. Don't hold the leads with your hands,you will get false readings.You can do the wires going back to main that way to. Make sure power is off. Just curious did it rain a lot lately, could water have gotten in some where? Did You say you ran romex in PVC conduit? You have to disconnect each building individuality if they are on the same circuit.

I saw the romex in conduit also.

Not suppose to do that!

He said he is using direct burial cable in pvc. The romex is above ground in the buildings. Again terminology, using the general term romex to describe both types of non metallic sheathed cable. Here we have NMd (dry) and NMw (wet) which are similar to US types.
It appears he has more than one building fed by a single breaker in a subpanel in another building.
You could use an ohm meter if it turns out he has only a little leakage to ground (suggested by the 60 sec interval to trip. Not a dead short but he still needs to do some disconnecting to isolate the fault to a part of this large and rather unusual circuit.

While there are multiple buildings on the one circuit, I should add that one building has a total of one duplex plug that powers an automatic sprinkler system, while another building has one duplex plug and an overhead light fixture with a 60-watt bulb (a woodshed). The barn has the majority of the outlets (I think there are 10, and two of those are controlled by switches to power fluorescent lights). So, loads aren't very high.

To answer the question about rain, we did have a couple of fairly heavy rains here earlier this season, but certainly not the heaviest we've had since the circuits were installed. Unless the underground conduit has been damaged, I don't think any water could have entered where the branch lines from the sub panel are contained.

max 12 devices per circuit is Ok but still not recommended.. Same procedure to isolate fault I explained earlier.

So disconnect the barn,It seem to have the most load first and see if it trips, then move on to the next one, after all are disconnected and it still trips has to be the feeders.

I'll keep you posted on progress...