First, you cannot move a mirror, that has a rest mass faster then light to replace the source of light. but let's disregard that, and say that there is a photon bouncing between two perfect mirrors. This is called a photon clock.
Now photons are scattered elastically, that is called Rayleigh scattering froma mirror, that is the only way to keep the photons' energy and phases and build a mirror image. Now in the case of elastic scattering, the energy of the photon is kept, so in the case of a perfect mirror, this can go on forever. Of course there are no perfect mirrors, so after a while the photon will be absorbed or inelastically scattered and loses energy or gives all its energy to the absorbing atom's electron.
When the photon is scattered elastically off an atom of a mirror, it's wavefunction, that describes its probability desctribution for all of space, will change by changing the speed vector's direction of the photon. Everything else, all the other characteristics of the photon are kept, this is the only way to build a mirror image.
So your question, whether the photon is slowing down or not, it is not. Photons always travel at speed c in vacuum, when measured locally. When the scattering happens, the photon's speed vector's direction is changed, but the magnitude is always c. That means, that the speed of the photon is always c before, during, and after the scattering process, what changes is just the direction of the speed vector. (This is true for absorption re-emission too). If there would be an absorption, re-emission, the photon would seize to exist, until that moment, its speed is c, and after the emission, its speed is c instantaneously.
After the comments, it is important to note that whenever the photon bounces off inside the mirror, it exerts pressure (momentum) on the wall of the mirror.