I connected the powerlines (both positive voltage and ground) on both ends of the little strip of development board that holds the shift registers and the AND-IC -- if power fluctuations towards the end of the 'train' were the problem, then that should have fixed it. But it didn't.
Then I disconnected all the connections from the third shift register. The shift registers share a common Output Enable, Clock and Strobe lines, all driven from a single pin from the PIC. If I disconnected the Clock and Strobe lines, but left the IC in the socket, the other two shift registers worked as intended!
I reconnected the Strobe line, and still all was well. Then I reconnected the Clock line, and things were defective again.
Apparently the voltages and/or the current on the Clock line was insufficient to drive three shift registers. I do drive that line straight off a pin of the PIC, but that was not a problem in the MACH F display.
So I changed the circuit: the pin from the PIC would drive a transistor, with the collector connected to the positive voltage 'power rail' (which sounds impressive, but it is really just a bunch of wire connections tied to the positive voltage from the wallwart adapter) -- this should ensure a correct voltage and as much current as the ICs want/need.
Alas, that did not work. I have absolutely no idea why.
So, I did what any engineer would have done: I worked around the problem. I still have a few pins left, so I connected one of the pins to the Clock pin of the third shift register. Instead of pulling one pin high or low, I now pull two pins high or low. And it works.
I'm not sure why -- but it works. And after ten days of frustration (sometimes I wonder why I consider this a hobby...), I'm not going to pick any nits regarding the solution.