I did measurements in the pA range using an electrometer. Usually, these sorts of measurements are done with a feedback ammeter. The other technique for measuring currents is the shunt ammeter(the voltage drop across a resistor). The box the OP pointed to is a shunt ammeter.
For the feedback ammeter, Ib and Vos are the dominant parameters. Vos is a strong function of temperature. I did design a 4-terminal I-V converter that had +-10V out in 4 decade ranges from 100 mA with bias and suppression.
If the is a uProcessor solution, you might be looking at an average current consumption. If the currents are tiny and you want an average, you can use coulomb meter and divide by time which I also did. The coulomb function is usually included in an electrometer.