We present numerical simulations of the evolution of low-mass, isothermal, molecular cores which are subjected to an increase in external pressure. If the external pressure increases very slowly, the core approaches instability quite quasi-statically. However, for faster compressions, a compression wave is driven into the core (Hennebelle, P., Whitworth, A., Gladwin, P. and André, P.: 2003a MNRAS 340, 870). Quantitative comparisons with observational velocity and density profiles are presented. The consequences of this compression for the fragmentation of the cloud is investigated and discussed.