How about using 28 guage flat ribbon cable with IDC sockets to plug onto header pins on the PCBs?
https://www.powerstream.com/Wire_Size.htm This wire guide says 28 guage can handle 226mA. If that's too little, just double up. Use a 10-pin cable and socket. The four wires on the PCB will each split into two pairs on the cable, and recombine on the other end.