Context. The kinematics and chemical composition of stellar populations of different ages provide crucial information on the evolution of the various components of a galaxy. Aim. Our aim is to determine the kinematics of individual stars as a function of age in IC 1613, a star-forming, gas-rich, and isolated dwarf galaxy of the Local Group (LG). Methods. We present results of a new spectroscopic survey of IC 1613 conducted with MUSE, an integral field spectrograph mounted on the Very Large Telescope. We extracted ∼2000 sources, from which we separated stellar objects for their subsequent spectral analysis. The quality of the dataset allowed us to obtain accurate classifications (T_eff to better than 500 K) and line-of-sight velocities (with average δ_v ∼ 7 km s⁻¹) for about 800 stars. Our sample includes not only red giant branch (RGB) and main sequence (MS) stars, but also a number of probable Be and C stars. We also obtained reliable metallicities (δ_[Fe/H] ∼ 0.25 dex) for about 300 RGB stars. Results. The kinematic analysis of IC 1613 revealed for the first time the presence of stellar rotation with high significance. We found general agreement with the rotation velocity of the neutral gas component. Examining the kinematics of stars as a function of broad age ranges, we find that the velocity dispersion increases as a function of age, with the behaviour being very clear in the outermost pointings, while the rotation-to-velocity dispersion support decreases. On timescales of < 1 Gyr, the stellar kinematics still follow very closely that of the neutral gas, while the two components decouple on longer timescales. The chemical analysis of the RGB stars revealed average properties comparable to other Local Group dwarf galaxies. We also provide a new estimation of the inclination angle using only independent stellar tracers. Conclusions. Our work provides the largest spectroscopic sample of an isolated LG dwarf galaxy. The results obtained seem to support the scenario in which the stars of a dwarf galaxy are born from a less turbulent gas over time.