For more permanent care, Leather Luster does work on these.
To clean them thoroughly when they're really filthy, take a can of Dow Scrubbing Bubbles, spray it on, and let it dissolve. The dirt will wash away, and if you need to, a follow-up with a squirt of water and a wipe will finish the job.
BTW, Scrubbing Bubbles do wonders on all my base shoes, too.
