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We consider the problem of planning collision-free motions for a mobile manipulator whose end-effector must travel along a given path. Algorithmic solutions are devised by adapting a technique developed for fixed-base redundant robots. In particular, we exploit the natural partition of generalized coordinates between the manipulator and the mobile base, whose nonholonomy is accounted for at the planning stage. The approach is based on the randomized generation of configurations that are compatible with the end-effector path constraint. The performance of the proposed algorithms is illustrated by several planning experiments.