Normally T-duality is introduced perturbatively by computing world-sheet spectrum of fundamental strings, and one of the conclusions is that it switches between momentum mode and winding mode of fundamental strings. My question is that does this switching extend beyond fundamental strings, namely, is it true that T-duality simply turns any momentum modes (for example, those carried by D-strings) into winding fundamental strings? If yes, is there a convenient way to see why is this the case?
Yes, whenever the momentum is conserved and T-duality holds, T-duality must map a conserved quantity such as this momentum to another conserved quantity, i.e. the string winding number in this case, and this fact is independent of the carrier of the momentum or the winding charge. In the general nonperturbative case, you shouldn't think about the charges as some "convoluted results of some particular behavior of some particular object" but as about conserved quantities generating symmetries that exist regardless of any spectrum of the objects.
This still leaves the question whether T-duality is valid nonperturbatively. In all the known simple enough supersymmetric string vacua where it is provable in perturbation theory, it also holds nonperturbatively. We can't rigorously prove this statement in the universal situation because we're lacking the universal nonperturbative definition of string theory. However, we may prove it for many vacua by various tools – argue it is true for vacua with the maximal supersymmetry which produce SUGRA with exceptional noncompact (in M-theory: discrete U-duality) symmetries (T-duality is a subgroup); we may prove it for IIA and $E_8\times E_8$ heterotic string theory using BFSS Matrix theory, and so on.