A generalized CUSUM procedure is constructed for sequential detection of a specified amount of change within a multi-parameter family of distributions when the initial value of the parameter is unknown. The stopping time of the procedure is defined on a doubly-indexed stochastic process whose asymptotics are studied in terms of its weak convergence properties when there is no change and when there is a contiguous change. Analogous results are also obtained for the Page-CUSUM procedure with known initial parameter. To account for misspecification of models, the results are derived for the case when the procedures are based on a form of distribution which is different from the true one. It is seen how the drift term which sets in after a change occurs, and drives the underlying stochastic process towards the decision boundary, slows down under model misspecification for both the Page-CUSUM and the generalized CUSUM procedures.