I think Angie is the white leather bag queen, so can probably give you the best advice.
I have a very pale gray leather bag, and it can be a little more tricky to keep clean, mainly I wipe it off with gentle baby wipes around the seams and the bottom. It's pebbled leather, but also very soft/flexible. I also have a white vegan bag, and have to do the same for it, but I can also use dawn or spray cleaners and it seems to pick up a lot less in the same time period as the leather.
I think maybe stiffer leather, or leather with a treated surface might pick up less. Also it's great if there are feet on the bottom of the bag if you have to put it down somewhere and you are careful wearing it with new dark denim, or new dark navy sweaters/coats -- those have been the iffy ones in terms of dye transfer for me.
ETA: I happily play the field with handbags in every season. I have eight I'm mostly using this spring, I tend to carry less black and less furry/fuzzy bags in the spring and summer, but otherwise they are pretty much year round. I have a wallet I transfer from bag to bag, and not that much else needs transferring, I add things for special outings of course.