In wikipedia the two concepts are related as(in Magnetostatics ):

$\vec j(\vec r)=\rho(\vec r)v$ where $v$ is the velocity with which the particles move.

I consider a volume in which initially there is no charge and initial time is $t\_0=0$ and $Q(t\_0)=0$
I tried to derive the above formula the following way:

$\frac{dQ}{dt}=I$

$\int dQ=\int I dt$

$Q=It$

$\rho(\vec r)V= I t$

$\rho(\vec r) \cdot A \cdot l= j(\vec r )\cdot A \cdot t$ ,where $A$ is the surface

$\rho(\vec r) \cdot \frac {l}{t}=j(\vec r )$

$\rho \cdot v=j(\vec r )$, where $v$ as said above, is the velocity of the charged particles.

If this derivation is correct, my question is:

Is the inital formula, only when the current density is not time dependent? Is there a more general formula for when $\vec j= \vec j(\vec r,t)$ ?