Photons actually move the same as any other particle does, it is momentum over mass for them too.

This framework provides a single unified description of motion that is equally applicable to photons and matter particles. Instead of focusing on the differences between particles, we are going to look where these particles are identical.

The motion of a photon is identical to the motion of all other particles.  Their momentum divided by their mass is the velocity they travel.  A photon momentum is hf/c and their mass is hf/c^2, so v= p/m = c.  This is why a photon always travel at c, and removes the mysticism that has often been attached to this idea. It doesn't travel at c because it travels at c, as only a rule.  It travels at c because that is the speed that it must travel with the momentum and mass it has from the energy it appears to have.

All particles travel at the speed that their momentum divided by their mass dictates.  The motion of a photon is independent of their 4th dimension vector because its momentum and mass from that vector in the time dimension is always in balance to produce a c motion.  This makes their energy vector in the 4th dimension independent of their motion in 3D space.

In matter, the velocity of motion is also just momentum divided by their mass. First you need the total energy, E² = (pc)² + (mc²)², that lets you define the m_rel for a particle with rest mass,  and finally m_rel = E/c² into the equation v= p/m_rel. The motion of the matter appears different because the rest mass puts their momentum out of balance with their mass.  This has the effect to pin their motion in 3D space to their vector in the 4th dimension. But ultimately their velocity is simply that particles momentum over their mass, the same as a photon.  This makes their motion in space dependent on their energy vector in the 4th dimension.  

By focusing on what is the same in all particles we see that the energy vector, their motion in the 4th dimension, is identical in all particles and that the mass and momentum they have directly results in their motion, and that this motion follows directly from their momentum and mass.