1. When on the latest version of spigot

2. When a mob chain including a furnace minecart is teleported through a gate,

3. The furnace minecart splits from the chain

4. Its teleportation is cancelled

5. Movement is not restored.

This is partially intended; spigot does not support PushX/PushZ.
Not sure if the teleportation cancellation is part of that intended behaviour though.

We have the capacity to teleport the minecart; we just lack the ability to move it after teleporting;
Would it not make more sense to avoid splitting the entity, and instead teleport it (with no restoration of momentum thereafter?)

It is indeed part of the intended behavior, as it can create an infinite teleportation loop otherwise

Nitpick: this behaviour repros on paper.

Normal furnace minecarts teleport properly and restore their momentum. This behaviour only occurs when the furnace minecart has passengers.

Update: almost certainly an independent issue contained within #225.
Closed as duplicate.