Disclaimer: I’m part of the glTF format working group and an author of GLTFLoader. 
I believe you’ll generally find filesize and parse time to be better with glTF, which is optimized for transmission, does not assume an SDK for parsing, and has relatively few ways to represent the same thing. FBX has more complexity, for reasons that aren’t particularly helpful for runtime use on the web (more details).
An increasing number of tools have very good glTF support, and if you’re working with one of those I think you’ll find glTF to be more future-proof and better supported. Unfortunately Autodesk’s Maya and 3DS Max are still difficult to use with glTF. If you have a preference for particular tools, I’m glad to suggest a workflow (or point out current limitations).
The gist of what you’re trying to do here (a model containing multiple animations, where animations affect morph targets) is all supported in glTF. Here’s a glTF example with animated shape keys, but only one animation: three.js examples