Silicon-based ceramics (SiC, Si3N4) are promising materials systems for high-temperature structural applications in gas turbine engines. However, the silica layer that forms on these materials is susceptible to attack from water vapor present in combustion environments. To protect against this degradation, environmental barrier coatings (EBCs) have been developed to shield the underlying substrate and prevent degradation. Here we report on elastic and thermal properties, as well as internal stresses of candidate multilayer coatings, as measured in situ using microfocused high-energy X-rays in a transmission diffraction geometry. Doped aluminosilicate coatings were investigated for their stability on a SiC/SiC melt-infiltrated substrate. The coatings consisted of a Ba1-xSrxAl2Si2O8 topcoat with a mullite or mullite+SrAl2Si2O8 interlayer, and a silicon bond coat. A numerical model was used to compare the stress results with an ideal coating system. Experiments were carried out on as-sprayed and heat-treated samples in order to analyze the strain and phase evolution as a function of multilayer depth and temperature. The phase transformation of the topcoat promoted healing of cracks in the EBC and reduced stresses in the underlying layers and the addition of SAS to the interlayer reduced stresses in thermally cycled coatings, but did not stop cracks from forming.