Right now the way things are anyone can walk into a gun show and buy a gun as long as they can produce an ID and pay for it.
That's not completely true. The same laws that apply outside the gun show apply inside the gun show. That means if you're a federally licensed firearms dealer, you must do background checks both inside and outside of gun shows. If you're not a federally licensed firearms dealer (i.e. a regular guy who just wants to sell a gun), you're not legally obligated or even allowed to perform a background check in a gun show or outside of a gun show.
If someone wants to buy a specialty weapon like a high powered military assault rifle then they should have a special endorsement on their ID that says they've satisfied the government's requirements for that endorsement, of course they should still be run through the verification system.
Aside from the fact that true military assault rifles are on average less powerful than the average hunting rifle, we already have an arduous process for purchasing those weapons or any other specialty weapon.
In order to buy a machine gun in the US, you have to first live in a state that allows you to own them. Second, you’re only limited to guns built before 1986. Then you have to submit to a 3-4 month FBI background check, get written permission from your local chief law enforcement officer, submit a set of fingerprints to the BATF, and pay for a $200 tax stamp. And you have to do this for every machine gun, suppressor, short barreled rifle, and short barreled shotgun you buy. These weapons are registered with the BATF and subject to random inspections.
You can't buy those weapons at gun shows or even most gun stores. They have to be bought from Class III dealers which are few and far between in the world of gun stores. So this...
Many of the weapons bought at gun shows end up down in Mexico being used by the drug cartels or by street gangs in our urban centers.
... Is a lie.