water running sound after tap is turned off

Hi. Over the past couple weeks I've noticed a water running sound after any tap or water source is turned off downstairs. It sounds like the supply line is 'back-filling' the water expended - so it only 'runs' for a short time, then stops. I've looked at the full lengths of copper piping visible from the crawl space and see no evidence of any leaks. Any clues re. what's causing the running water sound? Thanks!

Is it a "running" sound or a series of "clicks"?

It's just a running water sound...