We have discovered that the latest update of Windows 10, dubbed the "Creators Update", can cause connectivity to these devices to no longer work. One workaround we've found to remedy this problem is described below:
1. Download this .bat file.
2. Right-click the .bat file and select "Run as administrator".
3. Restart your PC.
When the PC reboots, Windows Mobile Device Center and device connectivity should be working as normal again.