I used Mapbox Studio to create a style using a local tile source. I exported the source to get vector mbtiles.

Now how do I use these mbtiles and the style (which is still in Mapbox Studio) to actually host my slippy maps?

I can't host my data with Mapbox.

There are plenty of options to read mbtiles and serve maps but I can't figure out how to style those maps on the fly using my CartoCSS from Mapbox Studio.

CartoCSS would need to be processed into a Mapnik stylesheet in order to be applied to your vector tiles. For serving and styling your tiles, I would recommend checking out tilelive and tessera.

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