First, the photovoltaic effect is the creation of voltage and electric current in a material upon exposure to light.
You are talking about the electron being excited to a higher energy level, so I will write about the photoelectric effect too.
In the photoelectric effect, the electron is ejected out of the material, because the photon's energy will be enough to match the electron's work function.
Now in the photovoltaic effect, the electron is excited too, but remains inside the material.
These electrons in the material, in your case the solar cell, are not free. They are loosely bound to the nuclei of the atoms of the material. These electrons, when they interact with a photon, and absorb the photon, gain kinetic energy.
Because these electrons are loosely bound, they can travel from nucleus to nucleus, and when they do, they travel very slowly. This is called drift velocity.
The drift velocity is very slow, but because the electrons are densely packed inside the material, the speed of electricity is close to the speed of light.
You are correct, that the energy of the photon will be transformed into the kinetic energy of the electron, and so the electron moves to the next nucleus. Now that nucleus already has a loosely bound electron, so that electron will be kicked off (will gain kinetic energy), and will travel to the next nucleus.
You are asking whether it is the kinetic energy of the electron that is moving, or is it the electron itself. In your case, the electron that absorbs the photon, will move to the next nucleus. That next nucleus already has a loosely bound electron, that will be kicked off, will gain kinetic energy and will move to the next nucleus. So the answer is that electrons are moving along the material to the next nucleus. The kinetic energy of the electron (that absorbed the photon) will be transferred this way along the material. The charge carriers are the electrons, because each electron in the way will move to the next nucleus. This is called drift velocity.
So it is the electron that is moving to the next nucleus, but then the next electron (that was loosely bound to the next nucleus) moves to the next nucleus too. So the kinetic energy is transferred too. But the kinetic energy is transferred step by step, nucleus to nucleus. It is not the original electron. Each electron moves just to the next nucleus, and kicks off the electron that was there to the next nucleus.