In this paper, we study the asymptotic expansion of the partial derivatives of a sequence of linear conservative operators. We find sufficient conditions on the sequence that guarantee that such an expansion can be derived from the asymptotic formula for the nondifferentiated sequence. Our results are valid even if the operators of the sequence present difficult conservative properties different form the classical preservation of the usual convexities. We use our theorems to obtain the complete asymptotic expansions and Voronovskaja formulae for the partial derivatives of multivariate versions of the Meyer-König and Zeller operators and the Bleimann, Butzer and Hahn operators.