Here's a practical idea for quickly changing between harmonica tunings in the same harp, which I've been using for a long time. However, I don't know if it qualifies as "on the fly". If that means within the same song whilst playing, then not! But if it means to switch tunings between songs, it could qualify with an extra tweak.
Way waaaay back, not long after I first started playing in New Zealand in 1976, I started modifying my harps. Initially just retuning reeds, then in about 1979 creating my first 11-hole "Stretch-Harps", with Regular Breath Tuning. To assist with both retuning and the harmonica size extension, I found that the picture-hanging putty
Blu Tack was very helpful! Here's a bit of info and a photo of how I used it in those early days:
https://www.brendan-power.com/harpgeeks.php#Blu-Tack
It's such an effective product that it's still popular today, 45 years later! Once stuck on a reed,
Blu Tack stays there indefinitely until you want to remove it; in which case, just scrape it off. I soon realised this could allow me to have harps with different tunings. For example, if you tune the 5 draw on a C Richter harp up from F to F# (the major seventh in 2nd Position, for Country Tuning), then stick on some
Blu Tack to take it back down to the F, you now have the correct amount of reed-end weight to switch between the two tunings. Simply stick the small blob of
Blu Tack on the rivet pad when you want the F#, and replace it on the reed tip when you want the F. Amazingly, this can be done repeatedly - seemingly as often as you want! (maybe eventually it will need refreshing with a new blob, but I never reached that point).
This could apply anywhere else on the harp, including the 3 blow for the Richter/Paddy-Richter switch.
Since the amount is correct for good tuning, the
Blu Tack switch only takes seconds to swap over - easily possible between songs on a gig. The time limiting factor is the coverplate removal/replacement. However, if you swapped out the coverplate screws with magnets, that would be instant. With the strong small neodymium magnets we have these days, this is pretty easy to achieve.
My SlipSlider harps have cover magnets. You can see a photo of various models with magnets in the covers at 2:30 in this video:
https://youtu.be/2vdm9_RmFnU?si=A0PyLwDtGkObtIwH
So Dominick... A combination of
Blu Tack on the reeds with magnetically-attached covers should give you a pretty effective way of changing between tunings in 10 seconds or so, on the same harp. It's not exactly instant "on the fly" between licks, but certainly usable for situations where you want a particular tuning for a song and don't want to carry two harps.