I did a test upgrade for my community and overall it is ok. I'm planning in the next couple of days to do the live upgrade to 4.0.2.

However, the background tasks for the test upgrade are still running.  I used the manual option to start them:

This tool will run all the background processes until they are all complete. If you close the page before it has finished, the task will continue the processing until they are all complete

 Now, obviously I want to stop the background processes on the test upgrade, so I do not overload the server with background processes running on 2 sites in the same time - for the test upgrade and for the live upgrade.

How I can do that? How do I stop all background processes for my test site (without deleting/crashing the test site obviously)? 

Disable the Queue task under Advanced Configuration -> View Tasks.