Within a circuit quantum electrodynamics architecture, we theoretically investigate the detection of a single propagating microwave photon traveling through a resonant microwave cavity dispersively interacting with a double quantum dot tunnel coupled to a lead. Under suitable conditions, a single photon in the cavity can induce a measurable change in the electronic occupation of the charge states. We develop a quantum cascade approach that enables a time-resolved description of a single-photon wave packet impinging on the cavity. We make use of a simple model of charge detector to assess the efficiency of our photo-detection configuration as functions of key parameters such as coupling strength, tunneling rate, temperature, and photon resonance linewidth. We finally highlight a measurement-induced backaction effect on the cavity mode associated with the dispersive, nonabsorptive detection process.