Context: Gaia parallaxes and photometric measurements open a three-dimensional (3D) era for the Milky Way, including its interstellar (IS) matter. Three-dimensional Galactic dust distributions are constructed in various ways, based on Gaia data and photometric or spectroscopic surveys. Aims: The assignment of radial motions to IS dust structures seen in 3D, or 3D kinetic tomography, would be a valuable tool allowing one to connect the structures to emission lines of the associated gas, which are now measured at increasingly higher spectral and angular resolutions, and rich in information on physical and chemical processes. To this end, one of the potential techniques is to establish a link between dust clouds and Doppler velocities of absorption lines imprinted in stellar spectra by the gas associated with the dust. This requires a relatively close correlation between the absorber column and the dust opacity. We have investigated the link between the strength of interstellar K I absorption and the opacity of the dust in front of stars in the Taurus area, and we have tested the feasibility of assigning velocities to 3D dust clouds on the basis of K I absorption data. Methods: We have obtained high spectral resolution and high signal-to-noise spectra of 58 early-type stars in the direction of the Taurus, Perseus, and California molecular clouds. We have developed a new, dual interstellar and telluric profile-fitting technique to extract the interstellar K I λλ 7665, 7699 Å absorption lines from stellar spectra and applied it to the new data and to archived spectra of 58 additional targets. In parallel, we have updated 3D dust maps reconstructed through the inversion of individual stellar light extinctions. To do so, we supplemented the catalog of extinction estimates based on Gaia and 2MASS photometry with recently published extinction catalogs based on stellar spectroscopic surveys. We used the 3D map and the set of velocity components seen in absorption to assign radial velocities to the dust clouds distributed along their paths in the most consistent way. Results: We illustrate our profile-fitting technique and present the K I velocity structure of the dense ISM along the paths to all targets. As a validation test of the dust map, we show comparisons between distances to several reconstructed clouds with recent distance assignments based on different techniques. Target star extinctions estimated by integration in the 3D map are compared with their K I 7699 Å absorptions and the degree of correlation is found comparable to the one between the same K I line and the total hydrogen column for stars distributed over the sky that are part of a published high resolution survey. We show images of the updated dust distribution in a series of vertical planes in the Galactic longitude interval 150–182.5°  and our estimated assignments of radial velocities to the opaque regions. Most clearly defined K I absorptions may be assigned to a dense dust cloud between the Sun and the target star. It appeared relatively straightforward to find a velocity pattern consistent will all absorptions and ensuring coherence between adjacent lines of sight, at the exception of a few weak lines. We compare our results with recent determinations of the velocities of several clouds and find good agreement. These results demonstrate that the extinction-K I relationship is tight enough to allow one to link the radial velocity of the K I lines to the dust clouds seen in 3D and that their combination may be a valuable tool in building a 3D kinetic structure of the dense ISM. We discuss limitations and perspectives for this technique.