According to an interview with P-Nintendo.com, Mario Kart DS producer Hideki Konno revealed that only the racing modes will be available for play over Wi-Fi.

Puissance Nintendo: Will the Battle mode be playable in Wifi?

Hideki Konno: Unfortunately, the Battle mode will not be accessible in Wifi. It will be thus only one mode of race to 4 players.(local wireless)
The rest of the interview, in French, can be read here. A similar interview from Nintendo Europe can be read here.