I do it all on the bike. I blow snot rockets and have a bento box for when riders might be behind me and use the terry part of my glove to wipe the excess.

If it is any consolation my nose runs on the trainer and in the warm weather, just not as often and it also runs when I eat. I never can be without a tissue. <weird> but I have been like that as long as I could remember.

When I scuba dived it was suggested to take a sudefed before. My dh suggested I try that before an outdoor ride, but I have yet to remember to see if it would work.