It was easier for shows to move when the physical productions were made up of backdrops, wagons and a couple of set pieces, rather than the more physically complicated sets these days.
Shows moved for any number of reasons - the theatre was promised to another producer or production, a show nearing the end of its shelf life moving to a less desirable theatre or a show trying to move to a better theatre.
My Fair Lady spent most of its run at the Hellinger - almost six years, and then its final six months at the Broadhurst and Broadway. No idea how it could have fit into the snug Broadhurst, but it sure didn't stay long there,
It's a little more odd with Birdie. The Martin Beck (now the Hirschfeld) has always had the stigma of being on the wrong side of Eighth Avenue, but still, it was a better theatre than the 54th Street, which had a rep as the home of flop musicals and where old musicals went to die. Seems they wanted to get to the Shubert and probably had to wait for Take Me Along (the Shubert's previous tenant) to close.
Begin at the beginning and go on till you come to the end: then stop.