Context. GG Tauri A is the prototype of a young triple T Tauri star that is surrounded by a massive and extended Keplerian outer disk. The central cavity is not devoid of gas and dust and at least GG Tauri Aa exhibits its own disk of gas and dust emitting at millimeter wavelengths. Its observed properties make this source an ideal laboratory for investigating planet formation in young multiple solar-type stars. Aims. We used new ALMA ¹³CO and C¹⁸O(3–2) observations obtained at high angular resolution (~0.2″) together with previous CO(3–2) and (6–5) ALMA data and continuum maps at 1.3 and 0.8 mm in order to determine the gas properties (temperature, density, and kinematics) in the cavity and to a lesser extent in the outer disk. Methods. By deprojecting, we studied the radial and azimuthal gas distribution and its kinematics. We also applied a new method to improve the deconvolution of the CO data and in particular better quantify the emission from gas inside the cavity. We perform local and nonlocal thermodynamic equilibrium studies in order to determine the excitation conditions and relevant physical parameters inside the ring and in the central cavity. Results. Residual emission after removing a smooth-disk model indicates unresolved structures at our angular resolution, probably in the form of irregular rings or spirals. The outer disk is cold, with a temperature <20 K beyond 250 au that drops quickly (∝r⁻¹). The kinematics of the gas inside the cavity reveals infall motions at about 10% of the Keplerian speed. We derive the amount of gas in the cavity, and find that the brightest clumps, which contain about 10% of this mass, have kinetic temperatures 40−80 K, CO column densities of a few 10¹⁷ cm⁻², and H₂ densities around 10⁷ cm⁻³. Conclusions. Although the gas in the cavity is only a small fraction of the disk mass, the mass accretion rate throughout the cavity is comparable to or higher than the stellar accretion rate. It is accordingly sufficient to sustain the circumstellar disks on a long timescale.