  We obtain mixed $A_p$--$A_\infty$ estimates for a large family of multilinear
maximal and sparse operators. Operators from this family are known to control
for instance multilinear Calder\'on--Zygmund operators, square functions,
fractional integrals, and the bilinear Hilbert transform. Our results feature a
new multilinear version of the Fujii--Wilson $A_\infty$ characteristic that
allows us to recover the best known estimates in terms of the $A_p$
characteristic for dependent weights as a special case of the mixed
characteristic estimates for general tuples of weights.
