According to Tag 0E1V of Stacks, given a ring map $R \rightarrow R'$ and two $R$-modules $K,M$, there is a base change map $$ R\mathop{\mathrm{Hom}}\nolimits _ R(K, M) \otimes _ R^\mathbf {L} R' \longrightarrow R\mathop{\mathrm{Hom}}\nolimits _{R'}(K \otimes _ R^\mathbf {L} R', M \otimes _ R^\mathbf {L} R'). $$ Lemma 15.97.2 [0A6A] gives some criteria for this map to be an isomorphism. Are there any useful criteria for the map to be injective?
For example, with $R=M=\mathbb Z, K=R'=\mathbb Q$, it is not isomorphic, but it is injective ($0 \rightarrow \mathbb Q$).
EDIT I found this related question. But in the example above $\mathbb Q$ is not finitely generated as a module over $\mathbb Z$.