That is the key conclusion from a recent working paper (pdf, ungated) by Arindrajit Dube (UMass), Oeindrila Dube (NYU) and Omar Garcia Ponce (NYU):
Does access to arms promote violent crime? We exploit a natural experiment induced by the 2004 expiration of the U.S. Federal Assault Weapons Ban to examine how the subsequent exogenous increase in the availability of lethal weaponry affected violence in Mexico. The expiration relaxed the permissiveness of gun sales in border states such as Texas, Arizona and New Mexico, but not California, which retained a pre-existing state-level ban. Using mortality statistics over 2002-2006, we show that homicides, gun-related homicides and crime gun seizures increased differentially in Mexican municipios located closer to entry ports in these other border states, relative to entry ports in California. Our estimates suggest that the U.S. policy change caused at least 239 additional deaths annually in municipios near the border during post-2004 period. The results are robust to controls for drug trafficking, policing, unauthorized immigration, and economic conditions in U.S. border ports, as well as drug eradication, military enforcement, and trends in income and education in Mexican municipios. Our findings suggest that U.S. gun laws have exerted an unanticipated spillover on gun supply in Mexico, and this increase in arms has fueled rising violence south of the border.
Below is just one graph with raw data . There are many more graphs, tables, and maps in the paper.
This paper (pdf, ungated) by Luke Chicoine reaches a very similar conclusion with a different methodology:
In the four years following the expiration of the U.S. Federal Assault Weapons Ban (AWB), the homicide rate in Mexico increased 45 percent. Over the same period, over 60,000 firearms recovered in Mexico have been traced back to the U.S. A difference-in-difference approach is used to estimate the effect of the expiration of the AWB on homicide rates in Mexico; states with a strong pre-2005 drug cartel presence are defined as the treatment group. The baseline estimates suggest the expiration of the AWB is responsible for at least 16.4 percent of the increase in the homicide rate in Mexico between 2004 and 2008.