Different sourdough starter behavior based on starter composition?
Generally speaking, it doesn't matter that much.  The final recipe, technique, and dough composition are all likely have a lot more influence on the final bread result than the composition of your starter (as long as your starter is healthy).

Some people have suggested maintaining different composition starters for different tasks (or to match final dough flour type), but I've never seen any evidence that this produces a significant effect.

I tend to retain whole-grain starters when I am refrigerating them between uses, as whole-grain flour seems to be a slightly more resilient food.  (I've had a couple all-white flours go bad in the fridge, but I've never had a whole-grain starter go bad, even after not being used for a couple months.  That's just my own anecdotal experience, though.)  

If you really want to make sure your starter is ready for a different food in a recipe than it usually has, you can always start feeding it the kind of flour you'll use in the final dough for a few days prior to mixing the final dough.  Again, as long as your starter is healthy, it should work fine.  Some people claim this has a minor benefit; others say they've never seen a significant difference.