Hygrothermal simulations are of major importance for critical problems in building physics, such as the application of internal insulation in heritage buildings. These simulations require numerous material parameters that are challenging to determine. We present measurements of typical internal insulation materials, calcium-silicate and autoclaved aerated concrete, which we expose to a warm, humid climate on one side and a cold temperature on the other side. We measure the moisture gain over time and determine the moisture profile at experiment end. In an inverse modelling approach, the measurements are used to identify material parameters, in particular vapour conductivity and capillary conductivity as a function of moisture content. We found the measurements of crucial importance for the accurate determination of these parameters. When the parameters rely only on isothermal measurements such as the drying experiment, the model fails to predict the capillary condensation process. We demonstrate this on a dataset from another study with interior insulation subjected to changing boundary conditions. The model calibrated with capillary condensation data reliably reconstructs measurements while the drying-calibrated model drastically underestimates the moisture content.