Locally optimum rank tests for $ k $-sample slippage problems of location and scale parameters in the nonparametric situations were derived by Kakiuchi and Kimura [5]. In this paper we shall consider a general class of slippage rank tests for those problems and investigate their asymptotic powers. The asymptotic relative efficiencies of the slippage rank tests with respect to classical optimum slippage tests are also investigated and it is shown that the efficiencies are the same as those in two sample problems.