I do a lot of rock climbing, and do a finger specific warmup for it, otherwise I get popping and clicking in my fingers, just like you describe. I don't use heavy grippers like you describe, so maybe someone else will have a more relevant post.
I use grip putty like this. Some sort of physical therapy putty might also work. I usually squeeze it with my free hand as I'm driving to the climbing gym, sometimes for a fairly long time, 10-20 minutes, alternating hands as I'm driving. It may be more than is required, but it goes a long way to loosen up all the connective tissues in my fingers. Other than putty, any medium or light resistance squeezable object would do (once on a road trip I used a plush dog's chew toy I bought at a grocery store). They're just a lot of different moving parts in the fingers, and it takes me a good long while to get them all loosened up and moving properly, but I see big improvement when I do.
I also do 5 minutes of light cardio to get my blood flowing.