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.
Same problem here: plugging in gas dryer, with dryer off, immediately trips GFCI. GFCI is new; dryer is old. I doubt the "fluctuating current" argument: a washer, with its different cycle starts/stops, is going to generate more "fluctuating current" events than a gas dryer. (Also the dryer is off when I plug it in.) I thoroughly cleaned the inside of the dryer, removing all lint, and brushing dust off of all components. Still trips when plugged in. So I disconnected/isolated hot in the dryer's control panel: still tripped when plugged in. Then I also disconnected/isolated neutral in the dryer control panel: still tripped! New test: without the dryer, ran a test lead from ground in the outlet to the gas pipe connection: trip! Measured a small AC voltage (30mV) between outlet ground and gas pipe. Next step (in this case) is to talk to some people about grounding requirements for gas supply pipes.
Best Answer
Somehow you have lost the neutral to the basement outlet. One way I have found that will often fix this is not rely on the device (switches, receptacles) to carry the power on down the circuit. Put them all under a wire nut with a short pig tail for the device at that location. Good Luck, P.