Suppose $R \to T$ is a ring map such that $T$ is flat as an $R$-module. Then for $A$ an $R$-module, $C$ a $T$-module there is an isomorphism
$\text{Tor}^R_n(A,C) \simeq \text{Tor}^T_n(A \otimes_R T,C)$
This can be proved directly by choosing an $R$-projective resolution $P^\bullet \to A$ and following through with the homological algebra. A more 'sledgehammer' approach us to use the Grothendieck spectral sequence $E_2^{s,t} = \text{Tor}^{T}_s(\text{Tor}_t^R(A,T),C) \Rightarrow \text{Tor}^R_{s+t}(A,C),$
which collapses under the assumptions above to give the required isomorphism.
In the case that $T$ is not flat as an $R$-module is it possible to build a spectral sequence that abuts to $\text{Tor}^T_n(A \otimes_R T,C)$?