Let Cb(s) be the space of real-valued functions F, continuous and bounded or a separable Banach space S, and letbe the subspace of those functions which have bounded and continuous (Fréchet) derivatives up to order r on S. In the special case that S = H is a Hilbert space, Giné and León (1980) have shown that a sequence {Pn} of probability measures on the Bore1 u-algebra of His weakly convergent to some limit measure P on the Whole space Cb(H) iff it is so on the subspace(H) for some r≥1 The aim here i s to establish counterparts for nonsmooth Banach spaces, in particular for the space S = C[0,1] of continuous functions on [0,1]. The approach given hereallows one to equip the qualitative result with rates, expressed in terms of a suitable K-functional, connected with the tightness of {Pn} and Jackson-type inequalities. Applications are given to Donsker's weak invariance principle for independent and dependent random functions as well as to a central limit theorem for martingale difference sequences on Banach spaces.