The mottling gene is recessive, so Mottled x Black will just make Black offspring, all carrying a mottling gene.
White in Cochin bantams is recessive white, so it's hard to say what you might get because recessive white hides everything. Blue male to White female, for example, might make Blue and Black offspring all carrying recessive white, or it might make those same colors with another color leaking through in patches, like silver or red. With recessive white, it's hard to make any guarantees unless you know their genetic background.
Assuming nothing is hiding, though, and your White Cochins are just solid black underneath the recessive white gene, White crossed to Lavender would make all Black offspring carrying both white and lavender. Crossed to a Barred male, you should get Barred females and single-factor or dark barred males, all carrying recessive white.
