I found the following in a paper:
Let $(T_t,t\geq 1)$ be the following semigroup $T_t = [e^{-t}+(1-e^{-t})P]^{\otimes n}$ where $P$ is a probability measure, we know that $$[aQ+(1-a)P]^{\otimes n} = \sum_{|I|\subset n} a^{n-|I|} (1-a)^n P^{\otimes n} Q^{\otimes n-|I|}$$ consequently we have that $$T_t 1_A(y^n) = \mathbb{E}[1_A(Z^nIy^n)]$$ where $$(w^nIu^n)_i = \cases{v_i \text{ if } i\in I \\ u_i \text{ if } i\in I^c}.$$
Can someone guide me through the proper definitions (among which, the definition of a tensor product between operators/semigroups) that leads to the conclusions? I have no idea how to reach them despite extensive research.