I guess I'll answer my own question.  By default, the 9360 apparently does not recognize docks due to a security issue from earlier docks, but the fix is quite easy. Restart the computer and enter BIOS by pressing F2 repeatedly during startup until the Dell screen appears and then goes into BIOS. At this point you'll see the "Settings" screen. Scroll down to "System Configuration" Select "Dell Type-C Dock Configuration and check the "Always allow Dell Docks" - click "Apply" Select "Thunderbolt Adapter Configuration" and check the three top options - Technology Support, Boot Support and Pre-boot Modules. Change the security to "no security" if you are using your own docks only. Otherwise, you might want to choose "user authorization" but then you'll need to use the laptop keyboard to connect each time. - click "apply" and you're off to the races. Reboot, connect to the dock and all of the peripherals connected to the dock will work - in my case my headset via dongle, my keyboard/mouse via dongle and my speakerphone via USB cable.
My home setup is a WD-19TB connected to two Dell monitors. I have a Logi Bolt dongle for my Keyboard/mouse and a Poly dongle for my headset plugged into the dock. My personal XPS-13 9310 recognizes everything and works as intended - I leave the laptop shut, plug in the USB-C from the dock and it is off to the races. I can unplug that and plug in my work Latitude 7430 and all is well - they keyboard and headset work just fine - as if they were wired to the computer. ....but... when I plug in one of my two older 9360 (the older version of the 9310 - the Dell naming convention is not intuitive, but was the topic for a different post), it will not recognize the peripherals. The monitors work fine, but the only way to get the keyboard and headset dongles to work is to plug them directly into the 9360. Both 9360's are the same and neither recognize the peripherals plugged into the dock. I'm thinking (hoping?) that I've got some sort of setting wrong, but I can't figure it out for the life of me. FWIW, I have a second identical setup (except a WD-19TBS and bigger monitors - but the same Logi Keyboard/mouse and Poly headset models) and the 9360's don't connect there either, while the other two - and a Precision 5530 - connect just fine. The three personal computers (5530/9310/9360) are on the same version of Win 11 while the office Latitude is on Win 10. Since the docks are mounted to the back of the monitor, moving the dongles isn't the easiest thing to do and defeats one of the reasons for the dock. Yes, I could pair the keyboards via BT to the 9360's and do the same with the headsets, but that isn't ideal and BT doesn't work reliably in my office - there is something causing interference. I'm stumped - and so is my office help desk - although that isn't saying much.      
I'm doing something wrong, but what? WD-19TB and 9310 vs 9360