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Washington (CNN)-- A bill that would toughen laws on gun trafficking and straw purchases of firearms won approval on Thursday from the Senate Judiciary Committee.

The measure now goes to the full Senate for consideration. It is the first proposal of its kind to begin moving through the legislative process since the Connecticut school shootings in December that killed 20 first-graders.

President Barack Obama and top Democrats are seeking a package of tougher gun measures that is opposed by the influential National Rifle Association, many Republicans and some Democrats.

Polls show public support for expanded gun laws, and advocates have mounted a major campaign to pressure Congress to act in the aftermath of the Sandy Hook Elementary School massacre and other mass shootings in recent years.

The Senate panel, at its meeting, also considered other proposals, including a new ban on semi-automatic weapons modeled after military assault rifles.

It recessed on Thursday before voting on the other measures, however, as some members left to attend a classified meeting.

The legislation it approved would make it harder for Mexican drug cartels to get their hands on guns sold in the United States and would punish those buying firearms or ammunition for criminals, said the committee's chairman, Sen. Patrick Leahy, D-Vermont.

It would also stem the flow of guns purchased legally in one state from reaching people in other states where they are not legally allowed to own them, said Leahy, one of the bill's authors.

Members voted 11-7 to send the proposal to the full Senate, with Sen. Chuck Grassley of Iowa, the committee's top Republican, joining Democrats in support.

A separate proposal by Sen. Dianne Feinstein, D-California, to update the 1994 assault weapons ban that expired a decade later appeared headed for approval despite strong opposition by Republicans.

Leahy, who had expressed concerns about the new ban, said he would support sending it to the full Senate so the debate could continue.

However, Feinstein's proposal is considered unlikely to overcome an expected Republican filibuster in the Senate.

"It's not going to achieve the goal its sponsors believe it will," argued Sen. John Cornyn, R-Texas, at the committee meeting.

He noted a study that showed the previous ban, which existed for 10 years, did little to reduce firearms violence, asking "are we really going to pass another laws that has zero effect and pat ourselves on the back?"

Feinstein contended her proposal would prohibit the kind of weapon used by a lone gunman in the Connecticut shooting.

The intent of the measure, which also would limit ammunition magazines to 10 rounds, was to "dry up" the supply of such firepower, she said.

Opponents including the NRA say such a ban violates the constitutional right to bear arms, while Feinstein and other supporters that include law enforcement groups note the previous ban survived several legal challenges.

The 1994 ban, which she also sponsored, expired a decade later when Congress didn't renew it.

On another issue, Republicans on the panel have expressed support for expanding the scope of mental health information submitted to the federal background check system used by gun sellers.

However, they differ with Democrats on other aspects of expanding background checks. Bipartisan talks have failed so far to come up with a compromise that would address the "gun show loophole," which critics say provides an avenue for people who know they cannot pass a background check to obtain guns through private sales.

The proposal advanced on Thursday targeted straw purchases, which often are made by a person assisting someone who is unable to legally buy a gun or doesn't want the weapon traced back to his or her name.

It is currently a federal crime to knowingly purchase a firearm illegally. Those convicted could face 10 years in jail and a fine of up to $250,000. The proposal would expand the law to ensure the straw purchaser was criminally liable.

Leahy has said the measure would help prevent repeats of deadly bungles that occurred during the notorious "Operation Fast and Furious" by giving border patrol agents a solid legal framework for catching smugglers.

During "Fast and Furious," agents from the Bureau of Alcohol, Tobacco, Firearms and Explosives attempted to apprehend smugglers by allowing them to purchase guns with recorded serial numbers in Arizona and transport them toward the border with Mexico in hopes of intercepting them.

Agents lost track of nearly 2,000 firearms, some turning up at the scenes of killings in Mexico. Two rifles from the program were found at the site of a December 2010 gun battle in Arizona that killed U.S. Border Patrol agent Brian Terry.

It was not clear whether either of them was used to shoot Terry.

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