Maxwell's Equations are usually written as:
$$\vec{\nabla}\cdot\vec{E}=\rho/\epsilon_0,$$
$$\vec{\nabla}\cdot\vec{B}=0,$$
$$\vec{\nabla}\times \vec{E}=-\frac{\partial \vec{B}}{\partial t},$$
$$\vec{\nabla}\times \vec{B}=\mu_0\left(\vec{J}+\epsilon_0\frac{\partial \vec{E}}{\partial t}\right).$$
But the last two might be written more clearly as
$$\frac{\partial \vec{B}}{\partial t}=-\vec{\nabla}\times \vec{E}$$ and
$$\frac{\partial \vec{E}}{\partial t}=\frac{1}{\epsilon_0}\left(-\vec{J}+\frac{1}{\mu_0}\vec{\nabla}\times \vec{B}\right).$$
And then they clearly tell you how the fields change. And now we know what it takes for the fields to not change. An irrotational electric field causes the magnetic field to be steady in time. A $\vec{B}$ field whose curl is exactly balanced by current, produces a steady electric field. But this doesn't tell us what the fields are. For instance if a particle was at rest its electrostatic field could be irrotational, and there could be no magnetic field (or current), so all the fields are steady. But there could also be a wave travelling through space that hasn't yet reached the particle (so the particle stays at rest ... for now, until the wave gets to it). The locations of the particles and their motions don't by themselves tell us the fields.
For a particle at rest, there is a natural and very simple solution, an inverse square electric field, unchanging, and zero magnetic field. To another observer moving at a constant velocity, they see that charged particle moving at constant velocity. So the fields they see, are a very natural solution (of the many possible) to the equations for a charged particle moving at constant velocity.
That's the most natural solution, and you can compute it. If you try to use the causal equations I listed above, it turns into a chicken and egg problem, the fields now depend on what they were in the past. Which is reasonable, but where do you stop?
Edit for Sofia
In the frame where the charge is at rest at point $\vec{p}$, the charge density could be $\rho(\vec{r},t)=Q\delta^3(\vec{r}-\vec{p})$ and the current density definitely is $\vec{J}(\vec{r},t)=\vec{0}$. (If the charge is extended, $\rho(\vec{r},t)$ could equal $\frac{Q}{4\pi R^3/3}$ if $\left|\vec{r}-\vec{a}\right|<R$ and zero otherwise, as another example.) There are many possible electric fields, but $\vec{E}(\vec{r},t)=\frac{Q}{4\pi \epsilon_0 \left|\vec{r}-\vec{a}\right|^3}\left(\vec{r}-\vec{a}\right)$ (outside teh charge) is a natural and simple one. And again there are many possible $\vec{B}$ fields, but $\vec{B}(\vec{r},t)=\vec{0}$ is a natural and simple one. There the Lorentz-transformed versions of these $\vec{E}$, $\vec{B}$, $\rho$ and $\vec{J}$ are obviously natural and simple solutions, though there are still many. Just as there are many possible solutions for electric fields when there is no charge present. For example, you could have a plane wave in any direction, with any magnitude, as well as countless other solutions.
Edit for Agnivesh Singh
What I assume is that the fields obey Maxwell. The first two Maxwell (about the divergence) are constraints. At any time, those need to hold. The other two (with the time derivatives) are about how the fields evolve (so very relevant to your question). What's nice about all four, is that if the constraints hold at one time, and the fields evolve by the evolution equations, then the constraints will continue to hold. What's unfortunate is twofold. One, that they are about the total fields, not about the field due to this or the field due to that. Second, they don't include boundary conditions, so there is no unique solution until you specify boundary conditions.
If I didn't assume the fields obey Maxwell, then actually it would violate conservation of energy and momentum. But that's because we assign an energy and momentum density to the fields such their flux through vacuum is conserved and that their flux at charges and currents is exactly the the force and power exerted by the fields on the charges through the Lorentz Force Law. So it's a bit cheating, but if you'd rather take the energy density and momentum density of fields as given, then we need the fields to satisfy Maxwell to conserve energy and momentum.
I am more interested in the mechanism . My question was aimed to know how electric fields propagate ?
The fields evolve according the the equations I provided. If by propagation you mean energy and momentum transport, I think that's a different question (meaning a new question is needed, including the research stage, not an edit to this question). The biggest problem is again that there is no unique solution to Maxwell, even with no charges, there are many possible fields. Throw in a charge and there are again still many possible fields, and the equations don't specify a field as being due to a charge, they are just about the total field due to everything.
There are other difficulties to propagation. If you moved your charge to make a large field nearby and then watched that large deviation propagated, it literally is a source free field (like radiation) and so even if it propagates at $c$, you might object because you wanted to know if fields due to charges propagate at $c$. For that you'd have to trace the disturbance all the way back to the charge. And if your charge is a point charge, the fields themselves blow up there, so now you have infinities to deal with. You can try to make that work, but how convincing is it going to be in the end? And if you have an extended charge, then my example of a simple and natural field (due to the charge) isn't really convincing either since it won't look like a sphere in all frames. If it was a sphere in it's own rest frame it will look pancake like in a moving frame. If it is a sphere in the moving frame, it won't look like a sphere in it's own rest frame. You can try a fluid model, but energy and momentum conservation for a nonpoint charge actually fail unless you have something that holds the extended charge together, charged fluids actually physically spread out.
There is also a completely different way to see causality, which is to use Jefimenko's Equations:
$$ \vec{E}(\vec{P},t)=\iiint \frac{\left(c^2\rho(\vec{r},t_r)+c|\vec{P}-\vec{r}|\dot{\rho}(\vec{r},t_r)\right)(\vec{P}-\vec{r})-|\vec{P}-\vec{r}|^2\dot{\vec{J}}(\vec{r},t_r)}{c^2|\vec{P}-\vec{r}|^34\pi\epsilon_0}d^3\tau$$
$$ \vec{B}(\vec{P},t)=\frac{\mu_0}{4\pi}\iiint \frac{\left(c\vec{J}(\vec{r},t_r)+|\vec{P}-\vec{r}|\dot{\vec{J}}(\vec{r},t_r)\right)\times (\vec{P}-\vec{r})}{c|\vec{P}-\vec{r}|^3}d^3\tau.$$
However if you took those as a given, instead of Maxwell, you not only lose some solutions to Maxwell (like a primordial radiation field) but it also begs the question since the fields are explicitly calculated from the charge and currents in the past (at $t_r$) where $t_r=t-\frac{1}{c}\left|\vec{P}-\vec{r}\right|$.
I'm not expecting anything I wrote to make you completely happy. But maybe learning these things allow you to ask (or find and understand) new and more detailed questions to address your concerns. So hopefully you learned as much as possible based on how you phrased your question.
