From Reason:

Both proposals share the same problems as any other effort to expand the reach of background checks. First, the categories of prohibited buyers are irrationally and unfairly broad, encompassing millions of people who have never shown any violent tendencies, including cannabis consumers, unauthorized U.S. residents, people who have been convicted of nonviolent felonies, and anyone who has ever undergone mandatory psychiatric treatment because he was deemed suicidal.
Second, background checks are not an effective way to prevent mass shootings, since the vast majority of people who commit those crimes do not have disqualifying criminal or psychiatric records. Third, background checks, even if they are notionally “universal,” can be easily evaded by ordinary criminals, who can obtain weapons through straw buyers or the black market. Fourth, voluntary compliance is apt to be the exception rather than the rule, and enforcement will be difficult, if not impossible.