Context. The chemical evolution of asymptotic giant branch (AGB) stars is driven by repeated thermal pulses (TPs). The duration of a TP is only a few hundred years, whereas an inter-pulse period lasts 10(4) - 10(5) yr. Direct observations of TPs are hence unlikely. However, the detached shells seen in CO line emission that are formed as a result of a TP provide indirect constraints on the changes experienced by the star during the pulse. Aims. We aim to resolve the spatial and kinematic sub-structures in five detached-shell sources to provide detailed constraints for hydrodynamic models that describe the formation and evolution of the shells. Methods. We used observations of the (CO)-C-12 (1 - 0) emission towards five carbon-AGB stars with ALMA (Atacama Large Millimeter/submillimeter Array), including previously published observations of the carbon AGB star U Ant. The data have angular resolutions of 0 ''.3 to 1 '' and a velocity resolution of 0.3 km s(-1). This enabled us to quantify spatial and kinematic structures in the shells. Combining the ALMA data with single-dish observations of the (CO)-C-12 (1 - 0) to (CO)-C-12 (4 - 3) emission towards the sources, we used radiative transfer models to compare the observed structures with previous estimates of the shell masses and temperatures. Results. The observed emission is separated into two distinct components: a more coherent, bright outer shell and a more filamentary, fainter inner shell. The kinematic information shows that the inner sub-shells move at a higher velocity relative to the outer sub-shells. The observed sub-structures reveal a negative velocity gradient outwards across the detached shells, confirming the predictions from hydrodynamical models. However, the models do not predict a double-shell structure, and the CO emission likely only traces the inner and outer edges of the shell, implying a lack of CO in the middle layers of the detached shell. Previous estimates of the masses and temperatures are consistent with originating mainly from the brighter subshell, but the total shell masses are likely lower limits. Also, additional structures in the form of partial shells outside the detached shell around V644 Sco, arcs within the shell of R Scl, and a partially filled shell for DR Ser indicate a more complicated evolution of the shells and mass-loss process throughout the TP cycle than previously assumed. Conclusions. The observed spatial and kinematical splittings of the shells appear consistent with results from the hydrodynamical models, provided the CO emission does not trace the H-2 density distribution in the shell but rather traces the edges of the shells. The hydrodynamical models predict very different density profiles depending on the evolution of the shells and the different physical processes involved in the wind-wind interaction (e.g. heating and cooling processes). It is therefore not possible to constrain the total shell mass based on the CO observations alone. Additional features outside and inside the shells complicate the interpretation of the data. Complementary observations of, for example, CI as a dissociation product of CO would be necessary to understand the distribution of CO compared to H-2, in addition to new detailed hydrodynamical models of the pre-pulse, pulse, and post-pulse wind. Only a comprehensive combination of observations and models will allow us to constrain the evolution of the shells and the changes in the star during the thermal-pulse cycle.
