The propagation law of blast stress waves in jointed rock masses is closely related to the characteristics of the joint filling medium. The water content and thickness of the filling medium affect the propagation characteristics of the blast stress wave in the joint surface. Based on similarity theory, mortar-red clay-mortar composite specimens with different moisture contents (36%, 39%, 42%, and 45%) and medium thicknesses (3 mm, 6 mm, and 9 mm) were prepared with red clay as the filling medium. The dynamic uniaxial compression test of the specimen under different strain rates was carried out by using a variable cross-section split Hopkinson compression bar with a diameter of 50 mm. The effects of strain rate, moisture content, and medium thickness on the stress wave propagation characteristics, dynamic stress-strain curve, peak stress, crushing energy consumption density, and crushing characteristics of mortar-red clay-mortar composite specimens were analyzed. The results show that the blocking effect of joints on stress waves increases significantly with increasing filling medium thickness and water content. With increasing strain rate, the peak stress and energy of the composite specimen increase and there is an obvious strain rate effect. With the increase in the thickness and moisture content of the filling medium, the peak stress, dynamic elastic modulus, energy consumption density per unit volume, and crushing degree of the specimen gradually decrease and the nonlinear deformation is more obvious. There is a good linear relationship between the peak stress and the thickness and moisture content of the filling medium. The existence of joints greatly weakens the damage effect of stress waves on the specimens behind the joints, and with the increase in the thickness and water content of the filling medium, the degree of damage to the specimens behind the joint surface decreases under the same impact pressure and the weakening effect becomes increasingly significant.