What produces an electromagnetic field? A charge that moves, or a charge that changes over time? I've learnt an electromagnetic field is produced by moving charges, i.e. a current.  
Is it the case, or is it actually the fact that the charge is changing at a given location?  
I mean: imagine I have a charge $q_1$ located at a given point of the space. Then imagine the charge changes and becomes $q_2$, but without moving (this is probably impossible? but let's imagine). Does it produce an electromagnetic field?
Or, without "imagining" a charge changes over time:
Is the EM field produced by the movement of the charge, of by the fact the charge vanishes from its location?
 A: Regarding a change in the charge, when you say

this is probably impossible? but let's imagine

─ let's not. Electric charge is conserved, both globally and locally, i.e. it obeys the continuity equation
$$
\frac{\partial \rho}{\partial t}+\nabla\cdot\mathbf j=0,
$$
and this is absolutely critical for Maxwell's equations to be internally consistent. All changes in the charge distribution can ultimately be traced down to the physical transport of particles moving from one place to another.
If you want to "imagine" a world where the laws of physics are so different that the total charge enclosed inside a given surface can change without there being a corresponding flux of charge through the surface, then you're no longer describing electromagnetism as we know it, and it is completely pointless to speculate about how radiation might look like in such a theory.
A: 
I have a charge ... located at a given point of the space. Then imagine the charge changes ..., but without moving (this is probably impossible? ... Does it produce an electromagnetic field?

Elementary charges (electron, proton and their anti-particles) in unbounded state have unchangable charges and by this electric fields of certain size. 
Charges emit EM radiation in the case of accelerations. This acceleration can be a spiral path induced by a magnetic field or be this during the deflection from an electric field.
 
Synchrotron radiation                .............................................................Bremsstrahlung
But there is one more detail. During the approach to the nucleus the electron emits EM radiation, called spontaneous emission. The energy for the emission has to come from somewhere. Now your point of a changing electric field come into play. 
What if the electric fields of the electron and the proton inside an atom get weaker during the approach of the electron towards the nucleus? The excess energy wil be realized in the form of EM radiation.
