This paper is devoted to investigating the properties of multilinear conditions and $A_{(P⃗,q)}$ conditions, which are suitable for the study of multilinear operators on Lebesgue spaces. Some monotonicity properties of and $A_{(P⃗,q)}$ classes with respect to P⃗ and q are given, although these classes are not in general monotone with respect to the natural partial order. Equivalent characterizations of multilinear $A_{(P⃗,q)}$ classes in terms of the linear classes are established. These results essentially improve and extend the previous results.