We study slow-down effects for bubbles formed in a cosmological first-order phase transition (PT) focusing on deflagrations and hybrids, where the bubble wall is preceded by a shockwave of heated plasma. Slow-down has been observed in multi-bubble simulations together with a suppression of gravitational wave (GW) emission, mostly for slow walls. We study the impact of the shock waves on the wall velocity around percolation, by considering steady-state single-bubble solutions and incorporating the possible heating effects by two different mechanisms. First, we investigate the slow-down experienced by a bubble expanding into an impeding shockwave, where the temperature is higher than at nucleation, and the fluid is no longer at rest. Taking into account such heating and kinematic effects, we find that the most significant slow-down occurs for the fastest walls, and thus cannot explain the suppression of the GWs observed in the simulations. However, these effects are stronger for PTs with a sizeable change in degrees of freedom unlike what is usually implemented in simulations, suggesting that the degrees of freedom can be an important additional parameter for characterizing the GW spectrum. For the second slow-down mechanism, we study heated droplets of false vacuum that shrink towards the end of the PT. By implementing a suitable boundary condition motivated by energy conservation, we show how the droplet velocity, interpreted here as the late-time velocity of the bubble walls, can be predicted from the properties of the initial deflagration/hybrid, in remarkable agreement with numerical simulations. Droplets are found to shrink more slowly for stronger PTs and slower deflagrations, with mild dependence on the change of degrees of freedom. Such slow droplets naturally correlate with a suppression of GWs, while geometrical properties such as the shock width play an important role as well.