Turned out to be....

Warped rear brake rotor. After driving a few minutes and using the brakes a few times, problem got progressively worse. As I watch the bike from the rear while pushing, and finally paying attention to the catching sensation interval, I saw the rotor hitting the brake pad and creating the resistance. I removed the rear caliper and rode the bike (braking carefully) for 1/2 an hour with no problems.

Time for a new rotor and pads. Its an old bike, I will probably disassemble and clean the rear caliper, refresh brake fluids, and bleed system in the process.

In retrospect, the clicking sound might have been better described as a clacking, but who knows?

Thanks for all the input, which has been very helpful and educational even if not used with this particular problem. I'm sure it will come in handy down the road.

You guys are great!