There are many options for the switching. Usually the more important switches are those for the reference current(s). So in the Keithley 2000 and 2002 these are SD5400 mosfets. The choice of the switch also depends on where you are switching:
The switch at the integrator side of the resistor is likely more sensitive to charge injection, but have the advantage of operating all at the same voltage level (near ground). The switches also only have to work at a low voltage (this is why the 74HC4053 is possible even with a +-10 V reference.
The switches at the reference side should be less critical with charge injection, but have to work with a higher voltage (e.g. the full +- ref voltage), but the switches at the high and low side work at a different voltage level. This makes matching of the on resistance more difficult. So the tendency is to use lower resistance switches in this case.
Finding a good switch is more than just a datasheet hunt - there are additional parameters important usually not found in the datasheets. Another problem is that with parameters like leakage current the spread from typical parameters to tested limits is often very large.
An important parameter can be the speed of switching: the more often the switches are operated, the more important jitter gets. So variants with fast switching, like the 34401 need low jitter levels (preferably below 100 ps). Low jitter usually is easier with faster switches. Even the rather slow modulating Solatrons and the circuit shown by Mickle T below take quite some effort for fast switching of the gates.
In principle JFETs make very good switches with low leakage, but I have a slight problem in switching them fast. Also the choice of p channel Jfets is very limited and the on resistance is kind of high. So they are mainly an option for switching at the integrator level.
CMOS switches integrate MOSFET switches with gate drivers. The extra P-MOSFET part may not be needed and add some extra charge injection, but could also do some compensation of the charge injection. Using separate MOSFETs like the Keithley meters (2000,2002) do might get slightly better parameters, but the extra circuit can also add capacitive coupling and supply noise.
The Keithley 2002 and what it looks like the ADvantest/Japanese T6481 use current sources instead of just resistors for the reference. This kind of eliminates the on-resistance (any might this way allow for very small switches) of the reference switch and thus needs a low on_resistance switch for the signal path, as there is no more compensation.
In the old times there where also meters that used diode current steering (HP3455) and just 4000 series CMOS gates powered by the reference.
My personal favorite for switching at the integrator side is the 74HC4053 or as a possible upgrade option the ADG633. They are very simple to use and low voltage MOSFETs do have a good value for the figure of merit R_on * C_gate. Despite there really low price they could be still good performance. The low price of the HC4053 comes at the price of less testing - so one might have to do extra leakage testing. The 34401 shows that those switches seem to work quite well, even in a scheme with very frequent switching, where charge injection and jitter might be critical.
Having 3 SPDT switches in one chip is also nice, as it helps with compensation of the on resistance.