Has anyone else noticed that the same transition music from Leave it to Beaver was used on episodes of Mister Ed (at least the 1st season of Mister Ed anyway).

Yes, I've noticed that too in many eppisodes.

I haven't noticed it, but I'll be on the lookout. I do notice on Mr. Ed that the transition music is a slower, blue's version of the Mr. Ed theme song. That's done on a lot of shows.

I think they're back to airing first season episodes, so I will listen for sure.

I noticed the same music on both shows as well, usually to open the show after the opening credits, and it surprises me. The shows were on different networks (LITB was on ABC by the time Mr. Ed premiered), filmed at different studios, and produced by different production companies, so I can't figure out how they can both use the same music. Maybe the music was in the public domain by then, if it even is now.