To factorize the expression $m^{3}(x-2)-m(x-2)$, we first look for common factors in both terms. We notice that $m$ and $(x-2)$ are common factors. Thus, we can factor these out:

$$m^{3}(x-2)-m(x-2) = m(x-2)(m^{2}-1)$$

Next, we recognize that $m^{2}-1$ is a difference of squares, which can be further factorized into $(m+1)(m-1)$:

$$m(x-2)(m^{2}-1) = m(x-2)(m+1)(m-1)$$

Therefore, the factorized form of the given expression is:

$$\boxed{m(x-2)(m+1)(m-1)}$$