I seem to recall most of the MLVs became depot tractor units after the end of the boat trains, reclassified in the departmental series (931 without looking it up?). I feel sure Ramsgate and Strawberry Hill used them, although others might know better. I would think any such use would be on the juice, possibly in pairs (thus with two power bogies and reasonable power to weight); each unit had two 250hp traction motors. An MLV also had a vacuum exhauster, so vac stock could hauled. As Gwiwer makes clear, off the juice is another world altogether - contrary to editorial comments in a certain railway magazine over the years, an MLV could NOT happily trundle up and down all day on batteries with an EMU in tow. On the East Kent the green MLV sometimes takes the EPB on the batts, but only on special occasions, and then usually with diesel assistance part of the way. There has been talk of fitting a diesel generator in one of the MLVs.
In service days the idea of battery power was to get passenger luggage in the MLVs on to the unelectrified quays at Dover Western Docks and Folkestone (soon to be destroyed, if anyone wishes to object), and so alongside the waiting ferry. In normal service haulage off the juice would have been confined to the short-lived TLVs.
They certainly hauled the water wagons, either singly or in pairs, as detailed above and again (mostly) on the juice. I don't know, but I would imagine diesel fuel for the Dover shunters would have been tagged on to the continental freights, or perhaps delivered by road. However, the gronks went gronking up to Ashford some nights (a sound I remember well), so possibly they took fuel only at Ashford. I emphasise that this is speculation.