For the sake of keeping this somewhat brief, I'm going to stay somewhat surface level.
When a vesicle is endocytosed, it is called a primary endocytic vesicle.
When multiple primary endocytic vesicles fuse together, they form a larger entity called an early endosome.
Early endosomes can also be called "sorting endosomes" because different portions of the vesicle are sorted into different destinations, including these things called "recycling endosomes."
Transmembrane proteins in endosomes
In my experience, transmembrane proteins largely get sorted into these recycling endosomes and are then re-inserted into the cell membrane.
To my knowledge and from some cursory review on the topic, this is a common outcome for transmembrane proteins; they are eventually recycled and placed back into the cell membrane via exocytosis after being sorted into a recycling endosome.
I'm sure others can add more explanation and some of the alternative routes these proteins can take, but in the interest of brevity and clarity, I thought it best to just stick with what I've come to know as most common.
Also, if you'd like a helpful source, see if you can get access to the following paper through a school or library service: Elkin, S. R., Lakoduk, A. M., & Schmid, S. L. (2016). Endocytic pathways and endosomal trafficking: a primer. Wiener Medizinische Wochenschrift, 166(7-8), 196-204.
In a phenomenon that I think is really cool, some receptors are thought to actually keep functioning after they are endocytosed and form these things called "signalling endosomes" (search "signalling endosomes NGF and TrkA" to read more).