Should background checks on Immigrants be as thorough as background checks done on gun buyers?

Would the Left object to vetting immigrants more than, less than or equal to that of a background check performed on American citizens who seek to buy handguns?

Certainly The Left wouldn't object to the vetting of a small portion of immigrants a background check that would be anything less than, say...that of a background check vetting an American citizen when purchasing a handgun.  Would they?

Immigrant vetting / Handgun purchase vetting.

It is NOT EASY to legally obtain a hand gun.

...[you] will need to furnish a photo ID and fill out Form 4473.  [This form] confirms your status as a citizen who is qualified to own a firearm in the United States.

[.] [the]  Background checks are performed through the National Instant Criminal Background Check System (NICS) [.] and  easily available to the Bureau of Alcohol, Tobacco, Firearms, and Explosives (BATFE), if necessary.

If and when a gun store closes, these forms become the property of BATFE.

  Further, please SEE THIS SITE:

It is a class C felony for any person who is not a citizen of the United States to carry or possess any firearm, unless...