Quantum state transfer by propagating wave packets of electromagnetic radiation requires tunable couplings between the sending and receiving quantum systems and the propagation channel or waveguide. The highest fidelity of state transfer in experimental demonstrations so far has been in superconducting circuits. Here, the tunability always comes together with nonlinear interactions, arising from the same Josephson junctions that enable the tunability. The resulting nonlinear dynamics correlates the photon number and spatiotemporal degrees of freedom and leads to a multimode output state, for any multiphoton state. In this work, we study as a generic example the release of complex quantum states from a superconducting resonator, employing a flux tunable coupler to engineer and control the release process. We quantify the multimode character of the output state and discuss how to optimize the fidelity of a quantum state transfer process with this in mind.