Divergence of current density and charge density In magnetostatic the  continuity equation is:
$$\frac {\partial \rho}{\partial t} + \nabla \vec j(\vec r)=0$$.
In my script the following is said:
Since we are in magnetostatic $\frac {\partial \rho}{\partial t} =0$ and therefore $\nabla \vec j(\vec r)=0$. Now I understand that this $\nabla \vec j(\vec r)=0$ means that in the medium we are observing there are no sources of the electric current density. But at the same time, since a current density is present, there should be a region in which a charge is changing over time. If the current density is going towards a region that means we are having a sink there, and if it is going out of a place we are having a source. How would we have current without a change in the value of some charge over time? What am I understanding incorrectly ?
 A: The presence of a current does mean charge is moving around, but not necessarily that the charge density is changing anywhere. If $\nabla \cdot \vec{j} \neq \vec{0}$, then we'd have sinks and/or sources. Since it is zero, all we have is charge flowing around with more charge coming in to replace it.
Imagine, for example, there is an infinite line of people walking at a constant speed. Of course, everyone is changing places all the time. However, the density of people is always the same, for whenever someone walks forward, the person behind fills in the space that would be left open. It is the very same idea. While charges are moving around, there is always some more charge to fill in the places that would be left open, and charges are always leaving a spot where another charge is entering. Hence, charge does move around with the current, but no sources or sinks ever occur.
A similar analogy is to observe the flow of water on a bathtub, for example. While water can move around, it has no sinks nor sources.
A: If you let $\rho$ be the density of water instead of charge then an example would be a whirlpool. The density of water is constant throughout the whirlpool but the water itself is still moving. The same holds true for your case.
Note that we made a continuum approximation here. Similar to how fluid density is also an approximation; the fluid is actually made up of small particles. If you were to look at individual charges moving then you would see the density changing rapidly but if you have enough charges and if you squint a bit you can approximate this as a density field.
