We introduce the concept of multi-sensitivity with respect to a vector for a non-autonomous discrete system. We prove that for a periodic non-autonomous system on the closed unit interval, sensitivity is equivalent to strong multi-sensitivity and justify that the result need not be true if the system is not periodic. In addition, we study strong multi-sensitivity and N-sensitivity on non-autonomous systems induced by probability measure spaces. Moreover, we first prove that if fn converges to f uniformly, then strong multi-sensitivity (respectively, N-sensitivity) of the non-autonomous system does not coincide with that of (X, f). Then we give a sufficient condition such that non-autonomous system is strongly multi-sensitive (respectively, N-sensitive) if and only if f is so. Finally, we prove that if a non-autonomous system converges uniformly, then multi-transitivity and dense periodicity imply N-sensitivity.