ATLASGAL-selected high-mass clumps in the inner Galaxy: XI. Morphology and kinematics of warm inner envelopes

(Abridged) Massive stellar embryos are embedded in warm envelopes that provide mass reservoirs for the accretion process onto final stars. Feedback from star formation activities in return impacts the properties of the envelopes, offering us a unique opportunity to investigate star formation processes. We aim to characterise the warm envelopes of proto- or young stellar objects in different evolutionary stages based on the morphology and kinematics of the $^{13}$CO(6-5) emission and to examine their relations with star formation processes. Using the APEX telescope, we obtained maps of the mid-$J$ $^{13}$CO emission with an angular size of 80" x 80" towards 99 massive clumps from the ATLASGAL survey. Our maps are classified based on morphological complexities, and the radial structure of the emission is characterised for simple single-core sources. The velocity centroids of the emission are compared to small- and large-scale gas kinematics, aiming to shed light on the origin of envelope kinematics. $^{13}$CO(6-5) emission is detected towards sources in all stages of high-mass star formation, with a detection rate of 83% for the whole sample. The detection rate, line width, and line peak increase with evolution, and the line luminosity is strongly correlated with $L_{\mathrm{bol}}$ and $M_{\mathrm{clump}}$, indicating that the excitation of $^{13}$CO(6-5) emission is closely related to star formation processes. In addition, the radial distributions of the emission for single-core sources can be well fitted by power-law functions, suggesting a relatively simple envelope structure for many sources. As for the envelope kinematics, linear velocity gradients are common among the single-core sources. Our comparison of kinematics on different scales suggests that the origin of the linear velocity gradients in the warm envelopes is complex and unclear for many sources.