I have Moto G, USB debugging works, but very strange:
when I try to debug code, 10-15 seconds after running was stopped on a breakpoint, debugger is disconnecting.
Never seen this before, is there any reason for that behaviour?
No special option ware set in Developer options.
It can be a bad USB cable or bad contact in USB port (either on Moto side either on your PC side). Try to use another cable and/or USB port.