Straight into the meter chip. Ib ~= 1pA typical. That includes the clamp diodes.
In autoranging meter ASICs, they would use "double pole" switches on the die because the "on" resistance of CMOS switches is rather high. They would also use differential input to the meter circuitry (switching the current path separately from the measurement path-- basically a Kelvin connection). An ADC with differential inputs (or an instrumentation amplifier plus a single-ended ADC) could be used in an non-ASIC version.
Best regards, Spehro Pefhany