The problem occurred when I switched back to a glossy paper, and the printer changed back to the PK tank. The print came out with no black. I ran a print check and there was no PK ink coming out at all. I then called Epson and was told to run a cleaning cycle - which I did, but still no black. I was told to run another cleaning cycle and then saw a tiny bit of black on the print check. In all, it took 3 cleanings to get the PK back.
Is this normal for the 3880? Should I expect this problem to occur every time I switch out to MK and back to PK?
No it isn't normal and it does sound more like a cartridge/ink problem rather than nozzle problem. Could have been a bubble, could be a defective cartridge.
Also, since this is a new printer and the inks are fresh, I suspect that an air bubble occurred in the PK - as opposed to an ink blockage. If this is true, then is there some way - such as shaking or tapping the PK cartridge - to remove air? I've seen this done in a video on You Tube with a different printer cartridge.
You should avoid shaking the cartridges. In fact an over vigorous shaking could cause the problem you had. Once in a while, you can rock the cartridges to help keep the pigments suspended.