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The control of flying qubits is at the core of quantum networks. As often carried by single-photon fields, the flying-qubit control involves not only their logical states but also their shapes. In this paper, we explore a variety of flying-qubit control problems using a three-level atom with time-varying tunable couplings to two input-output channels. It is shown that one can tune the couplings of a $Λ$-type atom to distribute a single photon into the two channels with arbitrary shapes, or use a $V$-type atom to catch an arbitrary-shape distributed single photon. The $Λ$-type atom can also be designed to transfer a flying qubit from one channel to the other, with both the central frequency and the photon shape being converted. With a $Ξ$-type atom, one can use the tunable coupling to shape a pair of correlated photons via cascaded emission. In all cases, analytical formulas are derived for the coupling functions to fulfil these control tasks, and their physical limitations are discussed as well. These results provide useful control protocols for high-fidelity quantum information transmission over complex quantum networks.