Shoes for Supinators/Underpronators

Anyone have any experience with good shoes for walking/crosstraining geared towards people whose feet supinate?
I have some flat soled Vans shoes for Box Squatting and will usually DL in my socks, but am looking for a good pair of shoes to do a little speed / mobility work in.

So far I’ve figured out that neutral cushioning is generally the way to go, but it’s hard to find a cushioned shoe that doesn’t have a gigantic heel. While a heel might be nice for oly squatting, I’m thinking it’s a disaster for everday posture.

Thanks.

In case anyone is in a similar situation foot-wise, I found that the Nike Air Vomero works very well. I tend to have a narrower foot, so they’re perfect for me, the asics cumulus usually have a bigger forefoot and the New Balances are usually bigger all around.