This configuration is a bit odd. The reason why a GFCI outlet has a line and load side is that a single GFCI outlet can offer GFCI protection to any downstream outlets (load).
If I were in your situation, I would install a new GFCI outlet in the first location, and then new standard outlets (your region might require tamper-resistant outlets) in all the downstream locations. This should make troubleshooting simpler, plus you only have 1 GFCI to deal with.
Alternatively, if you do wish to install a GFCI in each location, they should not be wired to the previous outlets load side, rather they should be wired in parallel like standard outlets (either a pig-tail or using the second screw on the same side of the outlet).
If after correcting the installation it is still tripping, then you have a current leak somewhere. A GFCI compares the current on hot and the current on neutral, and if the two are different by a certain threshold, the breaker trips. It could be a device plugged in somewhere else on the circuit, a nicked wire, etc. Start by unplugging everything on the breaker. Plug something into the GFCI outlet - it shouldn't trip, if it does, it's a wiring problem or the outlet itself is bad (if you just replaced it, this shouldn't be the case). One-by-one, start plugging in other devices upstream and downstream of the outlet. When/if it trips, you have found your problematic device or outlet.
I discussed this concern with the manufacturer (Pass & Seymour by Legrand) and they were able to provide me with a clear answer. The model number is 1595W3PKCC4 - and per design, upon being energized for the first time, these GFCIs are supposed to trip to test themselves. All subsequent cycling of power is not supposed to cause the GFCI to trip. It was my mistake to not validate my observation before posting my question. I have since cycled the power to these devices and they remain in the untripped state upon power restoration - so all is good. Thanks to you folks for your responses. Pass & Seymour was very helpful.
Best Answer
It just means that a wire termination (where a wire meets a receptacle, wire nut or switch) failed somewhere from the overload. It failed in an arc-fault mode, meaning it burned out like a fuse then arced across the gap, until the arc self-extinguished. If it had not, you'd be seeing your insurance man in the morning.
The connection didn't short against another wire, so it didn't make a short circuit. That's why it didn't trip the breaker. Breakers are not "Magic Master Detect-All's". They make those; they're called dual-mode AFCI/GFCI breakers. The AFCI would have tripped from the arcing, stopping the arc in case it did not self-extinguish.
So now you need to go hunt down the bad connection, starting at the panel. Electrical circuits are wired like a tree, they can have any number of branches, but they never loop back to themselves. Most are wired linear like a vine. The problem is either at the last functioning outlet or the first bad outlet. The top candidates are: