My question is simple:
Given $f: \mathbb{R}^{n+m} \to \mathbb{R}$, $f\in C^{k}(\mathbb{R}^{n+m})$ , and $X \subset \mathbb{R}^{n}$. Write $f$ as $f(x_1, \ldots, x_n, t_1, \ldots t_m)$.
When is $$\partial^{\alpha} \int_X f \, d\mu = \int_X \partial^{\alpha}f \, d \mu?$$
That is, what are necessary and sufficient conditions for the differential operator to commute with Lebesgue integration? Integration here is w.r.t the first $n$ variables .
Here $$\partial^{\alpha} f = \frac{\partial^{\alpha} f}{\partial t_{1}^{\alpha_{1}} \partial t_{2}^{\alpha_{2}} \cdots \partial t_{m}^{\alpha_{m}}}$$ with $\sum \limits_{i={1}}^{m} \alpha_i = \alpha \leq k$.
You probably won't find any meaningful necessary condition other than "all occuring terms need to be well-defined". The problem is that even if the function is awfully irregular, the identity might hold "by accident".
The search for sufficient conditions on the other hand is a bit easier. Essentially you need to exchange the limit in the definition of the derivative with the integral. Lets assume that $m = 1$ and we differentiate with respect to $t$: $$F'(t) = \lim \limits_{k \to \infty} \frac{F(t + h_k) - F(t)}{h_k} = \lim \limits_{k \to \infty} \int_X \frac{f(x_1, \ldots, x_n, t + h_k) - f(x_1, \ldots, x_n, t)}{h_k} \; dx_1\ldots dx_n\\ = \lim \limits_{k \to \infty} \int_X f'(x_1, \ldots, x_n, \xi_k(x_1, \ldots, x_n)) \; dx_1\ldots dx_n$$
Where $h_n$ is any null-sequence and $\xi_k(x_1, \ldots, x_n)$ is the term that occurs in the mean-value theorem. If we can exchange the limit with the integral, we are done. The usual tool to interchange limits and integrals is Lebesgue's dominated convergence theorem.
For example, if $X$ is bounded [or more generally: has finite Lebesgue-measure] and the derivative of $f$ with respect to t is bounded, you can interchange the limit and the integral. This yields the desired result, that the derivation and integration can be interchanged.
If $X$ is unbounded, things get more complicated. You need to bound $|f(x_1, \ldots, f_x, \xi_k(x_1, \ldots, x_n))|$ with some sort of integrable function. This can be quite difficult, as $\xi_k(x_1, \ldots, x_n)$ can behave rather irregular, as you can in general only assume $|\xi_k - t| \le |h_k|$. If the derivative fluctuates much near infinity, it might happen that you can't find an integrable dominating function. If you want to find a sufficient condition in this case, you essentially need that $\frac{d}{dt} f$ is absolutely integrable and doesn't fluctuate much around infinity.
If you want to calculate higher derivatives, you essentially need to check the aforementioned assumptions for every "stage" of the derivation.
You've mentioned in the comments that you want to restrict this to the Riemann-integral, but there is not much point in doing this. If a function is absolutely Riemann-integrable, then it is Lebesgue-integrable and the integrals coincide. There are conditions on when you can interchange a limit and a Riemann-integral, but they are much more restrictive than the dominated convergence theorem.
The only case that is different between the Riemann- and Lebesgue-case is if you consider the improper integral of a function that is not absolutely integrable. These cases are relatively difficult and I'm not aware of any universal condition to interchange the differentiation and the integral. Basically these cases need to be treated separately depending on the function $f$.