Most dry lubes are either parafin wax based or use a light binder to retain teflon powder. In both cases the carrier solvents for these will allow the new lube to mix with the old and over time as the lubes go away and need to be refreshed then there will be less and less of the motor oil and more and more of the other stuff.
So bottom line is just give'r a wipe and wash it if there's grit and apply the new stuff.
For my own needs that involve riding a lot in the rain I just use motor oil or my beloved chain saw bar oil which is a lot thicker so it doesn't run. I tried some of the "dry" lubes years ago and found that unless used far more frequently than grease and oil that they didn't protect against rusting worth a damn. Depending on the local climate YMMV. But really if used sparingly there is nothing at all wrong with motor oil. It's a great lube that seems to suffer only from a stigma of being "too easy and too common". There's really very little, if anything, that the specialty stuff does any better.
My own favourite witch's brew is the thick chain saw bar oil mixed 50-50 with mineral spirits. Goes on thin so I can wipe away the excess and dries over a day or to so a nice thin film that doesn't run or wash away in the rain easily.