We investigate the propagation of MHD waves in a magnetised plasma in a weakly stratified atmosphere, representative of hot coronal loops. In most earlier studies, a time-independent equilibrium was considered. Here we abandon this restriction and allow the equilibrium to develop as a function of time. In particular, the background plasma is assumed to be cooling due to thermal conduction. The cooling is assumed to occur on a time scale greater than the characteristic travel times of the perturbations. We investigate the influence of cooling of the background plasma on the properties of magneto-acoustic waves. The MHD equations are reduced to a 1D system modelling magneto-acoustic modes propagating along a dynamically cooling coronal loop. A time-dependent dispersion relation that describes the propagation of the magneto-acoustic waves is derived using the WKB theory. An analytic solution for the time-dependent amplitude of waves is obtained, and the method of characteristics is used to find an approximate analytical solution. Numerical calculations of the analytically derived solutions are obtained to give further insight into the behaviour of the MHD waves in a system with a variable, time-dependent background. The results show that there is a strong damping of MHD waves and the damping also appears to be independent of the position along the loop. Studies of MHD wave behaviour in a time-dependent backgrounds seem to be a fundamental and very important next step in the development of MHD wave theory that is applicable to a wide range of situations in solar physics.