Even without knowing which concrete start position for the chess pieces on a 10x10 board you have in mind, it can be said that it's still a clear win for the chess pieces.
Let's assume at start the chess pieces somewhere on ranks 1 and 2, and the checkers pieces on the last ranks, and we call the squares with index like in chess, say a1, a2, ... until i10, j10
First phase for the chess side is to bring the Queen, perhaps through squares b3 or c4, on square i10, and capture Qi10xj10. The checkers side can do nothing against this, and cannot capture back.
Second phase is to manage to block the black squares on first rank with own pieces, say Ra1, Nc1, Ke1, Ng1 and Ri1. Chess pieces are not forced to capture, so they can block the first rank and prevent the checkers pieces to promote. The other chess pieces except the Qj10 can be sacrificed in a way that there occurs no danger for the pieces on the first rank.
The last phase is simply to repeat the moves Qj10-h10 and back Qh10-j10, until the checkers pieces have no more moves. For the checkers side, to be stalemated means a loss.
Frisian draughts is alot like checkers but unlike its cousin, this game of draughts is played on a 10 x 10 board and allows you to capture what's both infront or right next to the pieces, so for example if you put a pawn infront of the draught piece it's allowed to capture it.
So this would probably solve the issue of rooks or queens capturing checkers pieces with ease or chess pieces going to white tiles. Of course they can't capture a piece if there's something obstructing it (and there's also that rule of mandatory capture but were ignoring that).
So how well would they fare against chess pieces?