The Ebony was originally glued in with either fish-glue , hide-glue, or horn glue. That is why they fall out.
Larry Potterfield has a vid on YouTube but I can't remember if he talked about glue types or not. Ive been using black epoxy, it isn't original, and it may make future repair difficult if ever required but It won't fail. It also works well to fill in any gaps if your fit is less than perfect. The only problem is it will also try to creep into the grain of the walnut and make you use bad words, so make sure you seal the walnut so the line is nice and razor sharp. I start by cleaning up any defects in the inletting to get nice razor sharp lines, then sealing with a very thin layer of thin super glue, sand lightly , repeat.
Make sure ebony is sufficiently oversized to allow for shaping.
When the epoxy is at about the leather stage I remove the clamp and start cleaning up epoxy and start rough shaping the tip on the sander when the glue is sufficiently set.
Uncured epoxy is toxic, breathe as little as possible.