Master text frames are designed to have text placed into them, not to be brought to a live page.
That said, as long as you don’t do anything to the frame as far as size or location on the live page, moving it on the master should move it on the live page.
So, did you make even a tiny change to size or location?
I was pretty careful not to make any change to size or location on the child pages, since I've read that can mess up the connection. If Master pages weren't intended to be brought onto a live page, what is the proper way to create non-standard (e.g. non 2- or 4-column) layouts of text and images that can be repeated throughout a document? Is there a better way?
That's not what I said. I said the purpose of master text frames is for placing text.
Regardless of whether it's text or graphics, however, if you need to override master objects on a live page on every page, a master item is not called for. Without seeing your layout, it's nearly impossible for me to know what's going on but I will be dollars to donuts that you are in some way changing those text frames.
That said, it seems to me, you're working backwards. The layout should be firmly in place before doing this.