The "pilot wave" is the same as the multi-particle wavefunction in quantum mechanics, so it evolves according to the Schrodinger equation. Such a wavefunction is not a field in three-dimensional space, it is a field in 3N-dimensional space where N is the number of particles.
The equation of motion for the particles in three-dimensional space depends on the gradient of the complex phase in that 3N-dimensional space. This part is exactly the same as classical Hamilton-Jacobi theory, which is an alternative representation of the forces of classical mechanics. But building trajectories from the quantum-mechanical wavefunction adds an extra nonlocal force.
You can't send nonlocal signals in Bohmian mechanics because you can't do that in quantum mechanics, and Bohmian mechanics is just quantum mechanics plus particle trajectories. The reason you can't do it in quantum mechanics is because the nonlocal correlations it induces aren't strong enough, they are only strong enough to add a nonlocal extra to a local signal (see quantum teleportation, which requires a local signal to be performed).