RICHMOND — The Virginia Senate approved a bill Wednesday that would give judges and juries leeway to decide whether someone who shoves a police officer without causing injury deserves the same felony assault charge as someone who punches or stabs.

Republicans criticized the party-line vote as undermining law enforcement, but Democrats said it was a step toward delivering the criminal justice overhaul they have promised for the special General Assembly session that kicked off last week.

“For too long in this commonwealth, there have been cases where the punishment is disproportionate to the crime,” Sen. Jennifer L. McClellan (D-Richmond) said during floor debate. “That is part of what people are marching in the street and demanding change for.”

Democrats, who control both chambers of the legislature, have said that the past 2½ months of demonstrations over racial inequity, triggered by the killing of George Floyd by Minneapolis police in May, highlight the need to change laws regarding police training, practices and accountability.

Both the Senate and the House of Delegates are tackling ambitious agendas, but Wednesday was the first time a measure has gotten to the floor in either chamber since the special session convened Aug. 18. Several weeks of work on criminal justice, social equity and budget issues are yet to come, with senators meeting at a science museum and delegates meeting virtually as a precaution against spreading the novel coronavirus.

Though Republicans have signaled support for some criminal justice measures, such as areas of police training and accountability, the debate over the assault bill exposed fundamental differences.

“What in the world are we doing here?” asked Sen. John A. Cosgrove Jr. (R-Chesapeake). “Have you watched television for the last couple of weeks? Have you seen what our police officers are going through? And here we are with a bill that’s going to actually make it easier for someone to actually assault a police officer.”

The bill’s sponsor, Sen. Scott A. Surovell (D-Fairfax), tried to defuse the debate by explaining what the bill would not do: “It does not defund the police. It does not grant anyone the right to assault first responders,” he said.

Instead, the bill would remove a mandatory minimum six-month jail sentence for assaulting a police officer and would let a judge or jury consider reducing the charge from a felony to a misdemeanor if the charged person is mentally or physically disabled or if there was no physical harm to the officer.

Since the 1990s, Virginia has categorized any assault on law enforcement as a felony. That has led to felony charges in cases in which an autistic teenager resisted arrest, for instance, or an angry person threw food at a police officer, Surovell said.

“You often see these charges coupled with situations where an arrest becomes unnecessarily aggressive, and that happens a lot more than it should,” he said.

But a long parade of Republicans countered that the bill sends a discouraging message to police at a time when they are under enormous scrutiny and morale is low.

“That message to law enforcement is that we don’t care about you,” Sen. Ryan T. McDougle (R-Hanover) said. “And we’re sending a message to people that riot in our streets . . . that if you encounter a law enforcement officer you don’t need to be concerned because if you assault them it’s not as serious.”

Other Republicans said they saw merit in the bill but suggested taking it up during the regular session in January. Democrats said it was urgent to act now. The bill passed on a party-line vote of 21 to 15 and will shift over to the House for consideration.

More legislation aimed at overhauling police policies is advancing in both the House and Senate.

On Tuesday, a House committee endorsed a bill to establish an alert system for mental health crisis situations and “community care teams” to respond alongside police in localities across the state.

It also advanced a bill that would prohibit police departments from acquiring surplus military gear such as grenade launchers, combat-configured armored vehicles and armed drones, and would require police to get a waiver from the state before using such equipment.

That bill, sponsored by Del. Dan I. Helmer (D-Fairfax), would also ban law enforcement from using chemical agents and rubber bullets, which have been blamed for causing injuries during this summer’s street protests. Herndon Police Chief Maggie DeBoard cautioned the House committee on Tuesday that rubber bullets are an important tool for subduing armed suspects without resorting to standard ammunition.

The bill advanced on a party-line vote, with all Democrats in favor and Republicans against.

Those measures — along with others establishing training policies, minimum qualifications for law enforcement jobs and civilian review panels — are likely to be debated on the floor of the House by Friday.

In the Senate, an omnibus bill that covers a wide range of policing issues won approval from one committee last week but is being vetted for fiscal impact by the Senate Finance Committee before probably moving on to floor debate.

That bill, sponsored by Sen. Mamie E. Locke (D-Hampton), requires the state to adopt standards for police training and for decertifying problem officers; prohibits no-knock warrants, chokeholds and firing into a moving vehicle; and bans police departments from taking grants or equipment from the Defense Department.