The purpose of AF-C (continuous) is to focus between the shots, so the focus can be continuous. But…
The camera will not focus between each shot when electronic shutter is enabled either the ES mode (electronic shutter mode) or MS+ES modes (mechanical + electronic).
In burst mode, focus and exposure are fixed at the values for the shot in each burst.
— Fuji manual: shutter type
BTW, this also applies to all the other Fujifilm cameras, including: X100x, X-M1, X-Tx, X-Ax