Conditions on expressing magnetic field in terms of curl of current density Given a current density distribution $\mathbf J(\mathbf x)$ inside a closed bounded region $\Omega$, the magnetic field at any point $\mathbf y$ outside of $\Omega$ can be expressed as
$$
\begin{aligned}\mathbf B(\mathbf y)&=\frac{\mu_0}{4\pi}\int_\Omega\mathbf J(\mathbf x)\times\nabla_{\mathbf x}\frac{1}{|\mathbf x-\mathbf y|}d^3\mathbf x\\
&=\frac{\mu_0}{4\pi}\int_\Omega\left[\frac{1}{|\mathbf x-\mathbf y|}\nabla_{\mathbf x}\times\mathbf J(\mathbf x)-\nabla_{\mathbf x}\times\left(\frac{\mathbf J(\mathbf x)}{|\mathbf x-\mathbf y|}\right)\right]d^3\mathbf x\\
&=\frac{\mu_0}{4\pi}\int_\Omega\frac{1}{|\mathbf x-\mathbf y|}\nabla_{\mathbf x}\times\mathbf J(\mathbf x)d^3\mathbf x-\frac{\mu_0}{4\pi}\int_{\partial\Omega}\mathbf n(\mathbf x)\times\left(\frac{\mathbf J(\mathbf x)}{|\mathbf x-\mathbf y|}\right)d^2 S(\mathbf x)
\end{aligned}$$
where $\partial\Omega$ is the boundary of $\Omega$, $n(\mathbf x)$ is the unit normal of $\partial \Omega$ and $S(\mathbf x)$ is the area of the surface element. Now, if the current density $\mathbf J(\mathbf x)$ is zero at the boundary $\partial\Omega$ (this can be achieved by slightly enlarging $\Omega$ if $\mathbf J(\mathbf x)$ is not zero at $\partial\Omega$) we can then drop the second term on the last line. Now we simply have
$$
\begin{aligned}\mathbf B(\mathbf y)&=\frac{\mu_0}{4\pi}\int_\Omega\frac{1}{|\mathbf x-\mathbf y|}\nabla_{\mathbf x}\times\mathbf J(\mathbf x)d^3\mathbf x
\end{aligned}.$$
If the current density $\mathbf J(\mathbf x)$ is continuous and differentiable, the above conclusion should be correct. However, $\mathbf J(\mathbf x)$ might not be continuous in $\Omega$, e.g., infinite thin coils inside $\Omega$ carrying electrical current. Is the above derivation correct for $\mathbf J(\mathbf x)$ containing delta functions? What kind of singularities in $\mathbf J(\mathbf x)$ is permitted?
 A: It seems you might be assigning causality to Maxwell that isn’t there, and/or believing that the magnetic field contribution from curls around electric field come from or include currents, as only one term? Not sure.
Let’s start by clarifying a few basics about what Maxwell says and what is causal. Then mention and link to Jefimenko.


The four Maxwell Equations (five relationships) are best understood as:

1.Electric charges cause electric fields that converge/diverge at the charge: $$\nabla \cdot \vec{E} = \frac{\rho}{\varepsilon_0} $$
(In other words, charges attract/repel: $F_{1,2}= k_e \frac{q_1q_2 }{r^2}$)

2.Currents cause magnetic fields that curl around the current: $$\underbrace{\nabla {\times} \vec{B} = \mu_0 \vec{J}}~\text{ } ~(+  \mu_0\varepsilon_0 \frac{\partial \vec{E}}{\partial t})$$
(In other words, currents attract/repel: $F_{1,2}= \mu_0 \frac{\vec{I_1}\cdot\vec{I_2}L}{2\pi r}$)
*By considering propagation time, Jefimenko (below) derives all of electromagnetism from just the above two.

3.Magnetic field lines are always closed, with no sources or drains of field: $$\nabla \cdot \vec{B} =0 $$
(Quite straightforward until here.)

4.The electric field curls around changes in the magnetic field:
$$\nabla \times \vec{E} = \frac{-\partial \vec{B}}{\partial t}$$
This is a consequence, but during Maxwell is considered an additional relationship.

5.The magnetic field curls around changes in the electric field:
$$\underbrace{\nabla {\times} \vec{B} = \mu_0\varepsilon_0 \frac{\partial \vec{E}}{\partial t} }~\text{ } ~(+  \mu_0 \vec{J})$$
This is a consequence, but during Maxwell is considered an additional relationship.

From Jefimenko's equations we know that 4., 5. are not causal - not from the curl to the derivative nor vice versa. The terms are due to current variations affecting each field individually. Fields are tools. If fields (Maxwell) are used, the terms in 2 and 5 must both be included, even if they come from one object.
Time-varying application of this, Jefimenko Equations:
https://en.m.wikipedia.org/wiki/Jefimenko%27s_equations
A: Interesting observation. As you have stated, the second equation only valid when the boundary contains all the current distribution inside. But is this what you are asking? You should open this question for objections as well. 
