This work addresses the mathematical analysis - by means of asymptotic analysis - of a penalisation strategy for the full discretisation of elastic wave propagation problems in quasi-incompressible media that has been recently developed by the authors. We provide a convergence analysis of the solution of the continuous version of the penalised problem towards its formal limit when the penalisation parameter tends to infinity. Moreover, as a fundamental intermediate step we provide an asymptotic analysis of the convergence of solutions of quasi-incompressible problems towards solutions of purely incompressible problems when the incompressibility parameter tends to infinity. Finally, we further detail the regularity assumptions required to guarantee that the mentioned convergence holds.