Some but I'm guessing it is minimal. The countries from Mexico to the south have had unstable regimes and violence for a long time (probably because the Spanish institutions were exploitative and exacerbated differences between the landholders/other and Europeans/natives).

There have been lots of violent and terrorist movements from the 70s onward and even prior to that. Guns didn't help but the US is far from the only suppliers or cause.

Brazil is also a huge producer of guns. Given Venezuela's location at the edge of Central and South America, its hard to say where they would get most of their guns from.

People have been killing each other long before guns were invented, so I doubt very much it has any real effect. The "uncontrolled US weapons market?" Give me a break! The biggest seller of weapons to those countries is the US Government itself!

The scale of potential violence with guns vs. without guns should be obvious, so not commenting on that.

As for the US weapons flow to Mexico, there was a NPR article last year on the topic [1]. Wikipedia [2] mentions "many of the traceable weapons come from American weapons markets and festivals that do not have regulations for the buyers", which very much is the uncontrolled weapons market, right?

To be fair, the Wikipedia page also points to the other direction with "weapons [..] come from either government personnel who defect to the cartels, or are sourced from Central American black markets".

[1] http://www.npr.org/sections/thetwo-way/2016/01/12/462781469/...

[2] https://en.wikipedia.org/wiki/Mexican_Drug_War#Gun_origins