Change of variable technique for conditional distribution

This might be a naive question, but could someone please tell me how the change-of-variables technique applies to conditional cases? My intuition tells me that there is no difference.

The change-of-variables technique is usually formulated for non-conditional cases, like for finding the joint distribution of $$(g_1(X_1,X_2),g_2(X_1,X_2))$$ where the joint distribution of $$(X_1,X_2)$$ is known. My question is: is the technique the same if the conditional distribution $$(X_1,X_2|Y=y)$$ is known and we want to know the conditional distribution $$(g_1(X_1,X_2),g_2(X_1,X_2)|Y=y)$$?

$$Y$$ is a random variable that is not independent from $$(X_1,X_2)$$, it could be a function of $$(X_1,X_2)$$. My intuition tells me that $$y$$ would be treated as a constant and the technique would be the same.