Both. Moving charges generate traveling waves propagating in transverse directions. But they also generate evanescent waves in a longitudinal direction.
Since electron diffraction experiments can be explained both according to de Broglie's formula and classical EM when considering measurements in a macroscopic setting, and since both are linear (superposition applies), we can assume a basic compatibility exists between the two.
But I would want to be careful about saying "matter had a wave character". I don't recall anything of that flavor said by de Broglie. He said that there is a wave associated with an electron or particle. The wave arises from movement of the particle and disappears if there is no motion.