For the DMM there are 2 steps of "calibration" or automatic corrections that may or may not run in the background.
The more common part the auto zero function to correct for an offset error.
The 2nd point that was usually found in the older meters (e.g. HP3455 and Keithley 19x, Keithley 2001 ) is a correction for drift in the ADC gain and possibly amplifier gain. These meters from time to time or in every conversoin cycle also measure a stable reference voltage (e.g. the 7 V or a rather stable 2 V (e.g. 7 V/4 from PWM or charge pump).
Some of the newer meters like the HP3456, 3458 do no use the perodic check of the ADC gain. The 3458 only does this in the ACAL procedure that is called seprately.
The main point to get away with the extra regular gain calibration is from having very stable resistors, so that the gain does not drift that much.
It still makes sense to include the option to do an internal ADC gain calibration.
For the Multi-slope ADCs there is usually no secret extra linearity correction that is run in between or background. There may be some factory calibrations and maybe special tests to correct for some of the more prominent INL errors. This could be somerthing like seprate gain for the positive and negative side or maybe a small square term. The HP3458 ADC has a somewhat odd looking correction hardware for the zero - not sure how and if it is used at all, as it can be turned off too. My suspicion is that it would do more harm than good.
At least for my MS-ADC I don't need any extra numerical correction to get good INL. The only point is measuring the slow slope to fast slope ratio and auxiliary ADC scale in an extra mode. This does not have to repeated very often (more like 1 time or yearly) and it does not need extra hardware.
If one has control over the MS-ADC, there is also no problem adding waiting time.
Control over, you mean another switch ?
The MS-ADC does not need an extra switch to add a pause. By design there is a switch as part of the ADC to seprate the input from the integrator. If needed one can extend the rundown phase and add some waiting in the µs to ms range.
There is no real magic in the 3458 or other high performance multi-slope ADCs. Much is a careful design to avoid mistakes that lead to avoidable noise and INL. There are several designs that got to the 8 digit level: HP3458, Solartron 7081 (though a bit slow), R6581 (with some numerical corrections), Keithley 2002 (though barely 8 digit), Datron 1281 and follow up Fluke meters.
The DC front end of the 3458 is not that much different from the 3456, mainly better parts and a more sophisticated reduction of the switching spikes.
There are mainly 2 types of front end options: AZ switching all the way at the input (3458, R6581) with a 1 stage amplifier or some kind of chopper stabilized amplifier at the front and switching auto zero only after that (K2002, Datron1281) with 2 amplifier/buffer stages. The difficulty with the HP like 1 stage design is keeping the switching spike small. So the charge injection is more than a small detail.
The difficulty with the chopper stabilized way is keeping the bias current and current noise low. The lowest noise AZ OP-amps have a rather high bias. Switching of the chopper amplifier part is more frequent, but with the advantage that it is between 2 voltages that are very close together.