The authors study the Mosco- and Hausdorff-convergence of fixed point sets of nonexpansive multifunctions. As a by-product, they give a counterexample which gives a negative answer to a question raised by A. Canetti, G. Marino and P. Pietramala [Indian J. Math. 33, No. 3, 203-220 (1991; Zbl 0797.41022)].