Spherically symmetric, general-relativistic accretion is known as the Michel flow. The stationary integral transonic solutions for the Michel flow have been constructed for multi-component accretion described by an equation of state where the adiabatic index varies with the radial distance along which the streamlines are studied, and the corresponding phase portrait spanned by such a radial distance and the flow Mach number has been obtained. Using techniques from dynamical systems theory, the nature of the transonic points of the aforementioned flow has been classified. The steady-state flow has been perturbed to study the stability of the stationary solutions, and it has been found that such flows are stable under the (linear) radial perturbation. As a consequence of the stability analysis, the corresponding acoustic spacetime embedded within the accreting matter has been obtained, and the horizon of the metric of such sonic spacetime has been identified by constructing the causal structure with the help of the Carter-Penrose diagrams. In this way, the accreting black hole systems in the general relativistic setup have been investigated from various perspectives: astrophysical, dynamical-systems, and analogue-gravity phenomena.