<p>This article focusses on the dynamics of the well-known pressureless Cargo–LeRoux model. The pressureless Cargo–LeRoux model provides a simplified framework for understanding fluid dynamics by focussing on the motion of fluid particles without considering pressure effects. It represents the one-dimensional hyperbolic flow of fluid particles and is significant in cases where pressure effects are negligible. An extensive study of the symmetries within the governing system is presented. It includes the computation of several types of symmetries, such as classical, non-classical and non-classical potential symmetries. The symmetries obtained using non-classical and non-classical potential approaches are more generalised, and the non-classical symmetries include the classical symmetries in particular. Symmetry reduction is performed corresponding to the obtained symmetries, and a few of the physically relevant reduced differential equations are shown here. Consequently, several novel analytic solutions to the considered model were found in terms of the Lambert, trigonometric, inverse trigonometric, exponential, hyperbolic, inverse hyperbolic and hypergeometric functions. Lastly, conservation laws are constructed through Ibragimov’s approach.</p>