I can't say exactly how you could measure thermal emfs precisely in your application, but I can share my experience with various test leads and my 34401A.
First of all, if you're working at the 10V level, these probably won't matter unless you want to go sub-ppm. My understanding is that's one of the reasons high-end references are 10V.
All my data is taken with the meter on the 100mV scale at 10 PLC, read out over RS-232, optionally averaged, and plotted (volts / seconds).
First, a warmup plot. This is averaged 10 times (100 PLC effectively). The inputs are shorted with a short, multi-stranded copper wire, bent backwards and stuck directly inside the jacks.
You can see my meter needs about 3 hours to stabilize, and that's with the window closed and AC off IIRC.
With the window open, it looks like this:
So my first point is: if you want sub-uV stability you probably need climate control or at least a stable environment, and you have to let your equipment stabilize for a few hours, or at least take a long series of readings so you can gain some confidence that the readings are stable.
Perhaps an obvious point, but the meter may not stabilize to exactly zero with shorted inputs, and it may stabilize to a different reading if you're shorting the inputs with the cables you're later going to use for your measurements - you may need to take that into account.
Now on to thermal emf "measurements". My procedure is as follows:
* insert two identical leads in the meter jacks
* short the leads at the far end, keeping them together (minimizing loop area - some of my leads are twisted)
* let things stabilize for a few minutes
* grab the positive lead between my fingers right where it enters the meter (that would be the body of the banana jack in most cases), with a reasonably strong grip, and trying to avoid touching the other lead or the meter itself
* watch the graph and hold tight until it starts to level off
* release the lead and wait until the graph returns to the baseline
* repeat for the negative lead.
Example graph, this time taken without averaging (10PLC):
Of course you can do this with different test leads and learn about their thermoelectric behavior. Except for bare copper wire, I couldn't find any that are significantly better than the data above. Within my procedure I find that CuTe Pomonas, CuBe MultiContacts and brass MultiContacts behave about the same, at least in conjuntion with my 34401A input terminals. Remember, these are "low-emf" connectors, not "zero-emf"
So here my advice would be to wait at least 5 minutes after handling the equipment in order for the thermal transients to fade away. Combined with the others' advice to also take measurements with reversed polarity, I think this issue is manageable.
Note that at first I tried to take some measurements off the DMM screen; it is MUCH easier and clearer to use a graphing software.