Bicycle Mechanics - Need help with gears

I recently bought a vintage peugeot iserna its and 86. The bike is in great condition and the gears worked well everything did. I had to switch the tire out both front and rear as they were dry rotted. after I did this the rear gears automatically keep shifting down untill the chain pops off. I am guessing i have a problem with my derailer. if Anyone has some tips on how to fix the problem or knows what may be wrong please help me out I would love to get this bike out on the road.

AFTER you did this??

Is it possible you bent the rear derailleur hanger?

Are you SURE you seated the rear wheel perfectly in the rear dropouts??

Given the vintage, I suspect it doesn't have index shifting. Back then the levers used a friction clutch to resist the derailleur return springs.

If your problem is that you can shift when pulling the lever, and it returns when you let go, you need to tighten the clutch screw in the middle of the lever, and/or lube and re-assemble the lever properly so that the clutch holds.

I do have lever shifters. The wheel is all the way in the dropouts and the hanger is fine. Right now whenever i pedal the bike downshifts on its own untill the chain pops off and if i out the chain back on it just down shifts again. I did not have the problem before taking the tire off and putting it back on.
thanks, for the help i am still tinkering with it keep the suggestions coming

Don't forget to reset the inner and outer RD limit screws to keep the chain on the freewheel. But you'll still have to tighten the screw in the center of the levers to get the friction clutch to hold. It shouldn't need excess pressure, so take the lever apart and see if you still have the non-rotating washer that's keyed to the lever boss or body.

fixed thanks