Context. The characteristic variability of cool giants and supergiants is attributed to a combination of stellar pulsation and large-scale convective flows. Full 3D radiation-hydrodynamical modelling is an essential tool for understanding the nature of these dynamical processes. The parameter space in our 3D model grid of red giants has expanded in recent years. These models can provide many insights on the nature and properties of the pulsations, including the interplay between convection and pulsations.

Aims. We treat 3D dynamical models of asymptotic giant branch (AGB) stars and red supergiants (with current masses 1 M-circle dot <= M-* <= 12 M-circle dot) similar to observational data. We aim to explore the relation between stellar parameters and the properties of the self-excited pulsations.

Methods. Output from global 'star-in-a-box' models computed with the CO5BOLD radiation-hydrodynamics code were analysed, particularly in regards to the pulsation properties, to find possible correlations with input and emergent stellar parameters. The fast Fourier transform was applied to spherically averaged mass flows to identify possible radial pulsation periods beneath the photosphere of the modelled stars. Stellar parameters were investigated for correlations with the extracted pulsation periods.

Results. We find that the pulsation periods varied with the stellar parameters in good agreement with the current expectations. The pulsation periods follow Ritter's period-mean density relation well and our AGB models agree with period-luminosity relations derived from observations. A mass estimate formula was derived from the 3D models, relating the stellar mass to the fundamental mode pulsation period and the stellar radius.

Conclusions. While the non-linearity of the interplay between the self-excited pulsations and the self-consistent convection complicates analyses, the resulting correlations are in good agreement with respect to current theoretical and observational understandings.

Stars on the AGB can exhibit acoustic pulsation modes of different radial orders, along with non-radial modes. These pulsations are essential to the mass-loss process and influence the evolutionary pathways of AGB stars. P-L relations serve as a valuable diagnostic for understanding stellar evolution along the AGB. 3D RHD simulations provide a powerful tool for investigating pulsation phenomena driven by convective processes and their non-linear coupling with stellar oscillations. We investigate multi-mode pulsations in AGB stars using advanced 3D 'star-in-a-box' simulations with the CO5BOLD code. Signatures of these multi-mode pulsations were weak in our previous 3D models. Our focus is on identifying and characterising the various pulsation modes, examining their persistence and transitions, and comparing the results with 1D model predictions and observational data where applicable. We produced a new model grid comprising AGB stars with current masses of 0.7, 0.8, and 1M⊙. Fourier analysis was applied to dynamic, time-dependent quantities to extract dominant pulsation modes and their corresponding periods. Additionally, wavelet transforms were employed to identify mode-switching behaviour over time. The models successfully reproduce the P-L sequences found in AGB stars. Mode-switching phenomena are found in both the models and wavelet analyses of observational data, allowing us to infer similarities in the underlying pulsation dynamics. These 3D simulations highlight the natural emergence of multi-mode pulsations, including both radial and non-radial modes, driven by the self-consistent interplay of convection and oscillations. Our findings underscore the value of 3D RHD models in capturing the non-linear behaviour of AGB pulsations, providing insights into mode switching, envelope structures, and potential links to episodic mass-loss events.