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Generally, if multiple cargoes are loaded and segregated by hold a Bill of Lading will determine the quantity in each hold. That quantity is then used at the discharge port/ports. If the customer only wants a part of a hold then a draft survey is done before and after the partial discharge to determine the amount of cargo remaining on board and/or discharged.

Bills of Lading are either issued by the load facility by scale weight or by ship's draft survey. Bills of Lading are verified by the ships draft survey.

On the Algoway/Algorail we could transfer cargo A “door”was closed that stopped the cargo from dropping down the hopper and onto the boom belt The cargo would fall into the forward end of #1 hold Also the boom ended over #10 hatch Cargo could be run off the boom and into that hatch
This was done along with ballasting Mostly on the St Clair river and those multiple stone dock trips

They know how much cargo was in each cargo hold. They usually will just unload a couple cargo holds and take on ballast to compensate for that cargo then move to the next unload dock. They could also calculate the vessels displacement before and after unload to determine how much was discharged. But my experience was that we'd unload certain cargo holds and it was figured out during loading how much cargo was in each of those holds.

By the draft marks from loaded to whatever it is when partial unloaded

By the holds. When the PRT unloads at St. Clair they unload holds 1,3,5 and holds 2&4 go to Monroe.

When a ship drops part of its cargo at one port and then goes to a different port and finishes unloading, how do they know much each port had delivered?