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. Plus something into the GFCI outlet - it shouldn't trip, if it does, it's a wiring problem. Start plugging in other devices up and downstream of the outlet. When/if it trips, you have found your problematic device or outlet.