Any tricks to remove drivers from a (sticky) cabinet

I've had similar problems and found that, with some speakers, it's easiest to remove the binding post plate in the back and push the drivers out from the rear.  Usually, this will allow me to remove the woofer(s) and the I can reach my arm up through that cut-out and push out the other drivers as well

If the removal of the binding post doesn’t work, and you’re forced to trying to pry the driver out, go to an auto parts store and buy a set of auto interior trim removal tools. They’re usually about $15 and come with several sizes and shapes of plastic pry tools that are designed to not scratch car interior trim, door panels, leather, vinyl, etc. I would go that route to prevent marring the leather. 

you can use a small inside caliper, into screw hole, expand tips and pull from the screw holes, not the edge.

$12. be there tomorrow

or improvise to pull from the holes, not pry the edge

using alcohol, you need to position the speakers to control gravity, i.e. fluid flow. you can lay the cabinets on their back, so fluid won’t run down the face, get an artist’s little paint brush or ... and try a spritz of alcohol, start swishing a bit inside the screw holes, wait a while, then tilt to use gravity to control the flow of fluid, some from the outer edge in any small crevice, bit by bit...

When I upgraded the drivers in my Alon speakers years ago, Carl Marchisotto recommended I use a certain sized screw that was just bigger than the hole in the driver, which was screwed into the front mounting hole of the driver until it became engaged and then I was able to pull the driver out from the front, similar to what was recommended by @elliottbnewcombjr 

I suggest trying the recommended methods posted below before prying against the cabinet, which seems to be asking for trouble.

This video shows a good way to remove woofers from the front:

One way to make this slightly easier is to heat the speaker just a bit with a heat gun, but, I think the approach shown in the video is really the safest.  You have to find the right screw to fit the driver hole, but, that is the only tricky part to this method.