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In advanced quantum processors, quantum operations are increasingly processed along multiple in-sequence measurements that result in classical data and affect the rest of the computation. Because of the information gain of classical measurements, nonunitary dynamical processes can affect the system, which common quantum channel descriptions fail to describe faithfully. Quantum measurements are correctly treated by so-called quantum instruments, capturing both classical outputs and postmeasurement quantum states. Here we present a general recipe for characterizing quantum instruments and demonstrate its experimental implementation and analysis. Thereby the full dynamics of a quantum instrument can be captured, exhibiting details of the quantum dynamics that would be overlooked with standard techniques. For illustration, we apply our characterization technique to a quantum instrument used for …