The Quantum Approximate Optimization Algorithm and its generalization to Quantum Alternating Operator Ans"atze (QAOA) is a promising approach for applying quantum computers to challenging problems such as combinatorial optimization and computational chemistry. In this paper, we study the underlying mechanisms governing the behavior of QAOA circuits beyond shallow depth in the practically relevant setting of gradually varying unitaries. We use the discrete adiabatic theorem, which complements and generalizes the insights obtained from the continuous-time adiabatic theorem primarily considered in prior work. Our analysis explains some general properties that are conspicuously depicted in the recently introduced . For parameter sequences derived from continuous schedules (e.g. linear ramps), these diagrams capture the algorithm’s performance over different parameter sizes and circuit depths. Surprisingly, they have been observed to be qualitatively similar across different performance metrics and application domains. Our analysis explains this behavior as well as entails some unexpected results, such as connections between the eigenstates of the cost and mixer QAOA Hamiltonians changing based on parameter size and the possibility of reducing circuit depth without sacrificing performance.