Galaxy evolution is driven by spatially distributed processes with varying time-scales. Integral field spectroscopy provides spatially resolved information about these processes. Nevertheless, disentangling these processes, which are related to both the underlying stellar populations and the interstellar medium can be challenging. We present a case study on NGC 613, observed with MUSE (Multi-Unit Spectroscopic Explorer) for the TIMER (Time Inference with MUSE in Extragalactic Rings) project, a local barred galaxy, which shows several gas ionization mechanisms and is rich in both large and inner-scale stellar structures. We develop a set of steps to overcome fundamental problems in the modelling of emission lines with multiple components, together with the characterization of the stellar populations. That results in the disentanglement of the gas ionization mechanisms and kinematics, along with an optimal parametrization for star formation history recovery. Our analysis reveals evidence of gas inflows, which are associated with the bar dust lanes traced with Hubble Space Telescope. In addition, we show the gas kinematics in a central biconical outflow, which is aligned with a radio jet observed with Very Large Array. The emission line provides estimates of electron density, gas-phase metallicity, and the mass outflow rate, allowing us to distinguish intertwined ionization mechanisms and to identify a part of the multiphase gas cycle in NGC 613. It traces the gas kinematics from the bar lanes to inner scale gas reservoirs, where it can eventually trigger star formation or AGN activity, as observed in the outflow.