Open chaotic systems are expected to possess universal transport statistics and recently there have been many advances in understanding and obtaining expressions for their transport moments. However, when tunnel barriers are added, which represents the situation in more general experimental physical systems, much less is known about the behaviour of the moments. By incorporating tunnel barriers in the recursive semiclassical diagrammatic approach, we obtain the moment generating function of the transmission eigenvalues at leading and subleading orders. For reflection quantities, quantum mechanical tunnelling phases play an essential role and we introduce new structures to deal with them. This allows us to obtain the moment generating function of the reflection eigenvalues and the Wigner delay times at a leading order. Our semiclassical results are in complementary regimes to the leading order results derived from the random matrix theory expanding the range of theoretically known moments. As a further application, we derive to the leading order the density of states of Andreev billiards coupled to a superconductor through tunnel barriers.