Is this for AC or DC? I don't like the solution proposed by fiddleback, because it will be limited in speed, and will require constant relay switching, which depending on speed of reading would just not work and age the relays in the DMM.
34465A is great but it isn't that fast, depending on how fast you need it to be, it might not work. What is your speed, time, accuracy and resolution requirements? How long do you need to it for, and what is the interval?
Any digitizing DMM does already what you need, by digitizing voltage. The way I would do it, create a little power probe, to convert current into voltage externaly, amplify it, and switch input of the multimeter with a relay, externally between the voltage and the current to voltage input, control all of it via serial or USB via a computer.