Context. Nitrogen bearing species, such as NH₃, N₂H⁺, and their deuterated isotopologues show enhanced abundances in CO-depleted gas, and thus are perfect tracers of dense and cold gas in star-forming regions. The Taurus molecular cloud contains the long L1495 filament providing an excellent opportunity to study the process of star formation in filamentary environments. Aims. We study the kinematics of the dense gas of starless and protostellar cores traced by the N₂D⁺(2–1), N₂H⁺(1–0), DCO⁺(2–1), and H¹³CO⁺(1–0) transitions along the L1495 filament and the kinematic links between the cores and surrounding molecular cloud. Methods. We measured velocity dispersions, local and total velocity gradients, and estimate the specific angular momenta of 13 dense cores in the four transitions using on-the-fly observations with the IRAM 30-m antenna. To study a possible connection to the filament gas, we used the C¹⁸O(1–0) observations. Results. The velocity dispersions of all studied cores are mostly subsonic in all four transitions and are similar and almost constant dispersion across the cores in N₂D⁺(2–1) and N₂H⁺(1–0). A small fraction of the DCO⁺(2–1) and H¹³CO⁺(1–0) lines show transonic dispersion and exhibit a general increase in velocity dispersion with line intensity. All cores have velocity gradients (0.6–6.1 km s⁻¹ pc⁻¹), typical of dense cores in low-mass star-forming regions. All cores show similar velocity patterns in the different transitions, simple in isolated starless cores, and complex in protostellar cores and starless cores close to young stellar objects where gas motions can be affected by outflows. The large-scale velocity field traced by C¹⁸O(1–0) does not show any perturbation due to protostellar feedback and does not mimic the local variations seen in the other four tracers. Specific angular momentum J∕M varies in a range (0.6–21.0) × 10²⁰ cm² s⁻¹, which is similar to the results previously obtained for dense cores. The J∕M measured in N₂D⁺(2–1) is systematically lower than J∕M measured in DCO⁺(2–1) and H¹³CO⁺(1–0). Conclusions. All cores show similar properties along the 10 pc-long filament. N₂D⁺(2–1) shows the most centrally concentrated structure, followed by N₂H⁺(1–0) and DCO⁺(2–1), which show similar spatial extent, and H¹³CO⁺(1–0). The non-thermal contribution to the velocity dispersion increases from higher to lower density tracers. The change of magnitude and direction of the total velocity gradients depending on the tracer used indicates that internal motions change at different depths within the cloud. N₂D⁺ and N₂H⁺ show smaller gradients than the lower density tracers DCO⁺ and H¹³CO⁺, implying a loss of specific angular momentum at small scales. At the level of cloud-core transition, the core’s external envelope traced by DCO⁺ and H¹³CO⁺ is spinning up, which is consistent with conservation of angular momentum during core contraction. C¹⁸O traces the more extended cloud material whose kinematics is not affected by the presence of dense cores. The decrease in specific angular momentum towards the centres of the cores shows the importance of local magnetic fields to the small-scale dynamics of the cores. The random distributions of angles between the total velocity gradient and large-scale magnetic field suggests that magnetic fields may become important only in high density gas within dense cores.