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We introduce a circuit-QED architecture combining fixed-frequency qubits and microwave-driven couplers. In the appropriate frame, the drive parameters appear as tunable knobs enabling selective two-qubit coupling and coherent-error suppression. We moreover introduce a set of controlled-phase gates based on drive-amplitude and drive-frequency modulation. We develop a theoretical framework based on Floquet theory to model microwave-activated interactions with time-dependent drive parameters, which we also use for pulse shaping. We perform numerical simulations of the gate fidelity for realistic circuit parameters, and discuss the impact of drive-induced decoherence. We estimate average gate fidelities beyond $99.9\%$ for all-microwave controlled-phase operations with gate times in the range $50-120\,\mathrm{ns}$. These two-qubit gates can operate over a large drive-frequency bandwidth and in a broad range of circuit parameters, thereby improving extensibility. We address the frequency allocation problem for this architecture using perturbation theory, demonstrating that qubit, coupler and drive frequencies can be chosen such that undesired static and driven interactions remain bounded in a multi-qubit device. Our numerical methods are useful for describing the time-evolution of driven systems in the adiabatic limit, and are applicable to a wide variety of circuit-QED setups.