The study of prestellar cores is critical as they set the initial conditions in star formation and determine the final mass of the stellar object. To date, several hypotheses are describing their gravitational collapse. Deriving the dynamical model that fits both the observed dust and the gas emission from such cores is therefore of great importance. We perform detailed line analysis and modelling of H_2_D^+^ 1_10_-1_11_ and N_2_H^+^ 4-3 emission at 372GHz, using 2'x2' maps (JCMT).Our goal is to test the most prominent dynamical models by comparing the modelled gas kinematics and spatial distribution (H_2_D^+^ and N_2_H^+^) with observations towards four prestellar (L1544, L183, L694-2, L1517B) and one protostellar core (L1521f). We fit the line profiles at all offsets showing emission using single Gaussian distributions. We investigate how the line parameters (V_LSR_, FWHM and T_A_*) change with offset, to examine the velocity field, the degree of non-thermal contributions to the line broadening, and the distribution of the material in these cores. To assess the thermal broadening, we derive the average gas kinetic temperature towards all cores using the non-LTE radiative transfer code RADEX. We perform a more detailed non-LTEradiative transfer modelling using RATRAN, where we compare the predicted spatial distribution and line profiles of H_2_D^+^ and N_2_H^+^ with observations towards all cores. To do so, we adopt the physical structure for each core predicted by three different dynamical models taken from literature: Quasi-Equilibrium Bonnor-Ebert Sphere (QE-BES), Singular Isothermal Sphere (SIS), and Larson-Penston (LP) flow. In addition, we compare these results to those of a static sphere, whose density and temperature profiles are based on the observed dust continuum. Lastly, we constrain the abundance profiles of H_2_D^+^ and N_2_H^+^ towards each core. We find that variable non-thermal contributions (variations by a factor of 2.5) are required to explain the observed line width of both H_2_D^+^ and N_2_H^+^, while the non-thermal contributions are found to be 50% higher for N_2_H^+^. The RADEX modelling results in average core column densities of ~9x10^12^cm^-2^ for H_2_D+and N_2_H^+^. The LP flow seems to be the dynamical model that can reproduce the observed spatial distribution and line profiles of H_2_D^+^ on a global scale of prestellar cores, while the SIS model systematically and significantly overestimates the width of the line profiles and underestimates the line peak intensity. We find similar abundance profiles for the prestellar cores and the protostellar core. The typical abundances of H_2_D^+^ vary between 10^-9^-10^-10^ for the inner 5000au, and drop by about an order of magnitude for the outer regions of the core (2x10^-10^-6x10^-11^). In addition, a higher N_2_H^+^ abundance by about a factor of 4 compared to H_2_D^+^ is found towards the two cores with detected emission. The presence ofN_2_H^+^ 4-3 towards the protostellar core and towards one of the prestellar cores reflects the increasing densities as the core evolves. Our analysis provides an updated picture of the physical structure of prestellar cores. Although the dynamical models account for mass differences by up to a factor of 7, the velocity structure drives the shape of the line profiles, allowing for a robust comparison between the models. We find that the SIS model can be cleary excluded in explaining the gas emission towards the cores,but a larger sample is required to differentiate clearly between the LP flow, the QE-BES and the static models. All models of collapse underestimate the intensity of the gas emission by up to several factors towards the only protostellar core in our sample, indicating that different dynamics take place in different evolutionary core stages. If the LP model is confirmed towards a larger sample of prestellarcores, it would indicate that they may form by compression or accretion of gas from larger scales. If the QE-BES model is confirmed, it means that quasi hydrostatic cores can exist within turbulent ISM.