DLC costumes purchased for LittleBigPlanet and LittleBigPlanet 2 will be usable in LittleBigPlanet Karting.
The PS Blog has a few caveats, noting that it will take the developers some time to ensure every costume is fully compatible with the racer, and also that the Assassin’s Creed and April Fools Fragrance costumes won't be making the jump - probably due to licensing issues, I'd imagine.
Additionally, any compatible costumes purchased from hereon in will be automatically available across three titles in the series - LittleBigPlanet 2, LittleBigPlanet Vita and LittleBigPlanet Karting. The original LittleBigPlanet and the PSP version are not so blessed, alas.
Double Eleven's LittleBigPlanet Vita releases in September, while LittleBigPlanet Karting - a joint effort by Media Molecule and United Front Games - is due in November.