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Recent spectroscopic measurements on periodically driven superconducting qubits have revealed transition energies consistent with Floquet quasienergies, validating the Floquet framework for engineered quantum systems. Understanding how the effective quasienergies and associated quasi-periodic Floquet states respond to external fluctuations and environmental couplings is crucial for achieving precise control and robust operation of Floquet qubits. In this work, we leverage Floquet master equations and Floquet geometric theory to develop a theoretical framework that connects the driven frame quasienergy response to dephasing and relaxation processes. We employ complementary decompositions of the lab-frame Hamiltonian, one emphasizing the period-averaged dynamics and one emphasizing the sub-period micromotion, to characterize how the effective spectrum of the driven quantum system evolves under fluctuating drive parameters. Building on this foundation, we analyze the dephasing and relaxation dynamics, exploring the emergence of the dynamical protection under single- and multi-tone driving in superconducting quantum circuits. Our results establish experimentally accessible signatures for coherence optimization and control of driven superconducting quantum circuits in an open quantum system setting.