I spawned an endermite by throwing ender pearls, named it with a name-tag and then got it in a minecart. However, after a couple of minutes the endermite despawned.

Steps to recreate:
1. Spawn an endermite
2. Name it
3. Put it in a minecart and wait

Some Extra Info (Don't rlly know if it's necessary or not)

• When the endermite went into the minecart, I couldn't see it's name anymore when I placed my crosshair over it
• I'm sure an enderman didn't kill it because I was watching the minecart the whole time and the endermen couldn't reach it
• This was all done in survival