depends on the production but normally on feature sets there is a DIT handling the copies and also helping with the monitoring LUTs, making dailies and offline versions on the fly, sometimes also making LTO backups, etc.
It varies a lot depending on the specific movie but the very basic setup is a laptop, couple of external drives, card readers, UPS backup for all the setup, and a backup software which is able to do checksum verified file transfers, preferably one which can also automatically create good enough reports for prod department and post production.
If making only the file transfers+dailies etc. and not on the fly LUT+live grading+color management work the DIT can usually arrive to the set later than the rest of the crew, in our productions usually about 3-4 hours so that he/she already has something to transfer when the DIT set is up and running. The DIT should always have a separate car and power supply so that it is possible to transfer+transcode the last card when the rest of the crew is wrapping, with mediocre setups this may take so long that the last car out of the set has to wait the DIT to finish so it is not ideal at all and the DIT needs to set up on another location every day after the shoot (hotel room etc.)
One thing to note is that a screw up may cost anything from 20K upwards on pro sets (reshooting is expensive) so the DIT should be very careful and always triple check everything (probably will never be hired again if screwing up the material even one single time, even if it's not your fault)
So a very basic setup:
-laptop, usually macbookpro, or a "trash bin" MacPro with at least Thunderbolt and USB3 . some setups may need a separate thunderbolt-->PCIe box and add on cards like a SAS card
-necessary software, for example Shotput Pro or Silverstack . for smaller setups, Resolve may be enough for quick grading tests and for making dailies and offlines, though it needs lots of GPU power and is a bit unreliable program.
-separate hard drives for 3 physical copies (will be stored on different locations after the shooting day)
-card readers, preferably two or even three so that you can swap them if having problems or use two at the same time if having lots of material like high speed, multiple cameras, etc.
-fast RAID array /arrays if needed to reduce transfer times, especially for the first copy
-extra monitor if needed
-very good UPS to protect the setup from (always very frequent) power problems like over/under voltage, power failure, enables you to shut down the genny for couple of minutes for refill without affecting transfers, etc.
-good small and silent generator +lots of gasoline (always have your own genny if doing time consuming transfers)
-a separate car for the DIT only so that you can have your own wrap time and don't have to hurry when everyone else is wrapping
-backup laptop and one card reader +hard drive(s) (don't have to be your own, just make sure that there is another laptop existing on set at all times even if someone else is using it normally for other work)
so that if your setup fries, gets stolen etc. you can still transfer simple copies using for example simple Finder drag->drop