As pointed out in the comment, the question makes more sense if we consider converting random variables. As for your statement about transforming into normal distribution, I hope you're referring to something like Box-Cox (which is an approximate method), not feature normalization with $(x-\mu)/\sigma$, since it's not transforming to normal distribution.
A standard way for converting a RV from some distribution into another is using Inverse CDF method. Typically, many random number generators use this method to convert the uniform distribution into an arbitrary one. Sometimes, this might not be enough since we can't get analytical inverse of $F(x)$, as in normal RV, and other methods exist, e.g. Box-Muller for uniform(s) to normal(s) conversion.
When the transform accepts going in reverse direction, we can first convert $X_1$ to $U$, then $U$ to $X_2$.