This is not the Lorentz force, which describes electric and magnetic forces on electric charges, it is the force of a magnetic field on a magnetic dipole (here's the wiki page about it).
When the coil is activated the magnetic field magnetizes the ferromagnetic spheres in the same direction (this corresponds to the magnetic field aligning the electron spin dipoles in the atoms of the spheres). Now that they are magnetized the magnetic field applies a force to them and pulls them towards the coil.
Quantitatively, the magnetic field $\mathbf B$ induces a dipole moment $\mathbf m$ in the spheres which gives them a potential energy
$$U = - \mathbf m\cdot\mathbf B$$
The corresponding force is
$$\mathbf F = -\nabla U = \nabla(\mathbf m\cdot \mathbf B)$$
In words this (roughly) says the force pulls the spheres in the direction of increasing $\mathbf B$ field strength. I say roughly because that explanation ignores any increase of $\mathbf m$ of the spheres, but $\mathbf m$ will always be induced parallel to $\mathbf B$ so the overall effect remains the same: Since the magnetic field strength is larger closer to the coil, the force pulls the spheres towards the center of the coil. At the center the field strength is at a maximum magnitude which means $\mathbf m\cdot \mathbf B$ is at an extremum, i.e. $\nabla(\mathbf m\cdot \mathbf B) = 0$, which implies $\mathbf F = 0$.
A force of 0 at the center of the coil is why you see in the video that with the coil left on the spheres get stuck: Any displacement of a sphere from the center of the coil causes a force to pull it towards the center, so that the center is a stable equilibrium point of no net force.
Therefore in order for the coil to speed up the spheres as they pass, it has to be turned off before they reach the center, otherwise the field will pull them back in when they come out the other side.
The way it works on permanent magnets is similar, and has the same force equation
$$\mathbf F = \nabla(\mathbf m\cdot \mathbf B)$$
Generally permanent magnets will have a much larger dipole moment than induced magnets, so the force is larger. However problem is that with permanent magnets, the dipole moment $\mathbf m$ is not induced in the direction of $\mathbf B$, rather it depends on the orientation of the permanent north and south poles of the sphere. The misalignment of $\mathbf m$ and $\mathbf B$ results in a torque in addition to the force
$$\mathbf \tau = \mathbf m\times \mathbf B$$
which tries to keep the poles aligned with the field. This is bad for the accelerator: The spheres want to roll for efficient motion, but the field wants to keep their orientations fixed. This is probably what he meant in the description about permanent magnets having erratic behavior.
