So I then invested in a more expensive mat hoping it wouldn't crack up. It didn't crack. However, the wheels continue to make fairly deep impressions into the mat - not as bad as the carpet, but bad enough that it was a nuisance and I felt that I had to drag the chair out of the impressions just to move it.
Next, I invested in oversized casters, which made a big difference - until I rolled off the mat onto the carpet. Then I was back to square one.
I figured I need to come up with a better solution. I spent some time googling and found a few people saying they'd used furniture movers/sliders, but I couldn't find a step-by-step so I had to make up a few steps, but it worked out.
I hope it works for you.
Step 1: Create a Firm Foundation
They're perfect because they're round, solid and have a pre-drilled hole that comes in quite handy. The only challenge is that pre-drilled hole is filled with a lag screw that needs to be removed for our purposes. I found that channel lock pliers give the right leverage, but I'm sure vise grips would work equally well.
Simply unscrew and discard the lag screws.
Once I removed the lag screws, I replaced them with T-bolts, which I found in the specialty hardware section at Lowe's, and that fit perfect into the pre-drilled hole.