This is probably not directly related to DisplayLink, as we are just the video part of this dock, and this sounds like a general USB problem.
I see you have the USB 3.0 controller:
Fresco Logic xHCI (USB3) Controller FL1009 Series
Driver Provider: Fresco Logic
Driver Version: 3.5.100.0
Driver Date: 2013/03/07
I think these were quite early USB 3.0 host controllers. It's possible this is having problems when there are a number of high bandwidth USB3.0 devices on the bus. It sounds like the Host controller or hub in the dock are locking up.
Have you tried connecting the dock to a USB 2.0 port to bypass the Fresco host controller and see if that shows the same issue?
Wim
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