I've done some research on South American militaria, and it is a strange situation. While Chile ended up following French lines, most of the other countries went along Prussian/German lines. Brazil is the exception but the Portuguese influence could be part of the reason.
What I have found is that Argentina supplied a lot of sun helmets to the other countries, and I think Argentina may have made some of the leather helmets as well. They have a huge cattle industry so that would explain it.
But even today the caps, uniforms, etc. These all look like they were made by the same one or two firms. I have seen Chilean visor caps and they're identical to Argentine, just with different badges. It was confirmed that Argentina likely makes these and supplies the other nations.