Are wave fronts in double-slit experiments moving forward? I ask a question about "What is a wavefront?" which follows the question "What makes the radiation behind a slit coherent".
For the wavefront it was answered, that "In electromagnetics ...(that are) the points that are at the same height." Now I'm wondering, are the wavefronts moving?
 A: Of course they move. How could they not?
I suspect that what is confusing you is that you are mis-interperting what a bright fringe on the screen is. If you are imagining that a bright fringe represents a point of instantaneous high amplitude then you have the wrong idea: a bright fringe is a point of maximum optical energy delivery (power). The actual field at a bright fringe will rise and fall sinusoidaly, it simply does so with higher amplitude than at less bright areas; at perfectly dark fringes the amplitude of that oscillation goes to zero.
Power goes by the square of the amplitude.
Now, why don't you see the oscillation in a diffraction experiment? Because the frequency of visible light is order of $10^{15}\,\mathrm{Hz}$.

In the comments here and in other places you have often made analogies to surface waves on water, which is good because they are mathematically equivalent but seems to contribute to the misconception about how you see fringes. Keep in mind the time and space scales associated with light and the limitation of human visual apparatus (and of machines that we use to capture images).
A: 
It is obvious that moving wave fronts will made moving fringes on an observer screen, will they not?

If the detector screen is in the yz plane and the slits are at particular y values and the light is originally going in the x direction, and you had light polarized in the z direction then indeed the electric field hitting a single point on the screen will sometimes be in the $+\hat z$ direction and sometimes in the $-\hat z$ direction. And if you fix a point on the screen the variation it might be very very small in amplitude over time, or it might be large in amplitude over time.
What we call bright fringes are the places where the amplitude of the variation over time is large. They do not move. The place with large amplitude stays where it is. And the place with small amplitudes stays where it is.
Please reread dmckee's excellent answer.

This explanation is not right for water waves, is it?

It would be an entirely appropriate way to talk about water waves. Only one modification is needed. For the electromagnetic wave you can talk about the intensity (or the amplitude, they are related but different) at any point on the yz plane that is the detector screen in particular lots of different z locations have their own intensities.
In a water wave, the intensity can only vary in the y direction since the z direction is actually used to measure the amplitude.
The wave goes up and down over time (height of the water going up and down for the water, direction of the electric field going from $+A N/C$ to $-A N/C$ in the electromagnetic case). For each value of y there is an amplitude of the oscillation and a corresponding intensity. And that amplitude of oscillation could be large or small depending on your y value. The places (ys) were it is large are due to constructive interference. The places (ys)/were it is small are due to destructive  interference.
The same interference that happened away from the screen/detector, such as right in front of where the wave hits the screen.
