Context. Dense molecular filaments are ubiquituous in the interstellar medium, yet their internal physical conditions and formation mechanism remain debated. Aims. We study the kinematics and physical conditions in the Musca filament and the Chamaeleon-Musca complex to constrain the physics of filament formation. Methods. We produced CO(2-1) isotopologue maps with the APEX telescope that cut through the Musca filament. We further study a NANTEN2 $^{12}$CO(1-0) map of the Musca cloud and the HI emission of the Chamaeleon-Musca complex. Results. The Musca cloud contains multiple velocity components. Radiative transfer modelling of the CO emission indicates that the Musca filament consists of a cold ($sim$10 K), dense (n$_{H_2}sim$10$^4$ cm$^{-3}$) crest, which is best described with a cylindrical geometry. Connected to the crest, a separate gas component at T$sim$15 K and n$_{H_2}sim$10$^3$ cm$^{-3}$ is found, the so-called strands. The filament crest has a transverse velocity gradient that is linked to the kinematics of the nearby ambient cloud. Studying the large scale kinematics, we find coherence of the asymmetric kinematics from the 50 pc HI cloud down to the Musca filament. We also report a strong [C$^{18}$O]/[$^{13}$CO] abundance drop by an order of magnitude from the filament crest to the strands over a distance $<$ 0.2 pc in a weak far-ultraviolet (FUV) field. Conclusions. The dense Musca filament crest is a long-lived (several crossing times), dynamic structure that can form stars in the near future because of continuous mass accretion. This mass accretion appears to be triggered by a HI cloud-cloud collision, which bends the magnetic field around dense filaments. This bending of the magnetic field is then responsible for the observed asymmetric accretion scenario of the Musca filament, which is, for instance, seen as a V-shape in the position-velocity (PV) diagram.
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