The DMM6500 and DAQ6510 operate the same way, in general.
The front panels of each will operate like a standard DMM with single input to a single A/D converter.
The sample rate, determined by several settings, will govern the measurement speed on each channel (either front panel or at each channel on a switch card).
The scan rate will determine how quickly each channel will change/switch. This only governs the channel switching, not the measurement itself.
To achieve a desired 'total rate' you would need to account for the sample rate of the measurement and the scan rate of the switch card/relays.
Multiplexers operate in a sequential switching configuration, so you would follow a sequence like the one below:
CH1 close + measure > CH1 open > CH2 close + measure > CH2 open > CH3 close + measure >......
Here, you'll find the scan rate will impact the open-close portion of the operation and the sample/measure rate will impact the measurement once the channel is closed/selected.
You will need to refer to the datasheets for each of the instruments and the switch card to determine specific rate specifications:
https://www.tek.com/en/datasheet/dmm6500-6-1-2-digit-bench-system-digital-multimeter-datasheet
https://www.tek.com/en/datasheet/daq6510-data-acquisition-and-logging-multimeter-system-datasheet
The instrument uses an analog input for the signal from your DUT.
It will then be processed through an A/D converter for reporting as a reading.