Disordered quantum trajectories describe repeated measurements in which the instrument applied at each step is chosen by an external environment. We present a limit-theorem framework for such trajectories at the level of general quantum instruments on a finite-dimensional system, allowing perfect, imperfect, and standard-Borel outcome spaces. After constructing a measurable density realization of the instrument family, we define the quenched laws of both the measurement record and the posterior-state process.

The central assumption is a uniform block Doeblin minorization for the quenched posterior-state kernels. Under this condition, the pullback dynamics on posterior laws is exponentially contracting. Consequently, there exists a unique equivariant family of random posterior laws, and its barycenter is the unique dynamically stationary state for the associated non-selective channel cocycle. When the environment is ergodic, time averages of posterior states converge almost surely, under the quenched law, to the annealed stationary state.

We then prove quenched fluctuation results for bounded additive observables along the posterior-state chain, including variance asymptotics, a central limit theorem, and Berry--Esseen bounds. Under an additional summable mixing assumption on the instrument process, the Berry--Esseen estimate holds with deterministic normalization. We also show that the stationary annealed state-path process inherits the mixing of the instrument environment up to an exponentially decaying error, thereby giving a route to further annealed and nonconventional limit theorems.

Finally, we discuss finite-outcome measurement records as applications. For perfect measurements, a coupling/admissibility mechanism based on Kraus-induced label dynamics transfers annealed central limit theorems for finite pattern counts from the dynamically stationary initial state to arbitrary random initial states. For imperfect measurements, where a visible outcome may correspond to several unresolved microscopic branches, the same transfer is recovered through an augmented block Doeblin condition on the joint law of the posterior state and the observed outcome block. Thus, empirical word frequencies in disordered measurement records have Gaussian fluctuations with the stationary centering and asymptotic variance, while the underlying method is the posterior-kernel Doeblin framework for disordered quantum trajectories.