Herman Allen’s hand shook as he held up a photograph of his year-old grandson, Joseph Allen, who died in a fire at an unlicensed day-care center last year.

The home ignited and the toddler died of smoke inhalation, strapped to a car seat in an upstairs room.

“His lungs burned up,” Allen said. The caregiver was unlicensed, and the family soon learned that she had lied to parents about how many children were in her home.

After heart-wrenching testimony from Allen and other relatives of children abused or killed in day care, state lawmakers moved forward Tuesday with legislation to better regulate child-care centers in Virginia.

Legislators have stepped forward with proposals meant to make child care safer in response to a Washington Post investigation into deaths in day-care operations. The investigation, published in October, found that 60 children had died in day care in Virginia since 2004.

The strongest bill, a measure put forward by Sen. Barbara A. Favola (D-Arlington) that would require all day-care operators to be licensed, failed in a Senate subcommittee. But several other proposals moved forward, including a requirement for fingerprint background checks for caregivers, a licensing requirement for state-subsidized centers, a ban on employing child abusers or sex offenders, and more rigorous checks with Child Protective Services.

September O’Brien of Loudoun spoke to the need for more aggressive interaction with CPS. A mother of three, she said her 4-year-old son Colin for months “begged me not to take him to pre-K.” But he wouldn’t say why. Finally, when he came home with rug burns and bruises all over his back, he revealed that a day-care worker had been dragging him across floors and over wooden furniture.

A CPS investigation was launched, but before a decision was made, the woman who abused the child had found a job as a director at another day-care center. Such centers are only required to do background checks when they first hire an employee, and the woman had not yet been placed on the state child abuse registry. A bill from Sen. Jennifer T. Wexton (D-Loudoun) would require annual background checks that include communication with CPS.

Sen. J. Chapman Petersen (D-Fairfax) said he was concerned about a “radical” expansion of licensing requirements that would force even occasional babysitters into a regulatory regime.

The lawmakers also waffled on whether to lower the threshold for licensure from six or more children in a home, and whether to include the caregiver’s own children in that total. Elly Lafkin, whose infant daughter Camden suffocated at a caregiver’s home in 2012, advocated for that change.

The woman watching her daughter had a grandchild and five other children in her home. Had she applied for a license, the state would have learned what Lafkin could not from a basic court records search: the caregiver had several aliases and a long list of criminal and civil violations.

“Under this law, she would have been out of business,” Lafkin said. She added that she found it “almost offensive” that state-subsidized centers would be subject to stronger regulations than others.

Still, she is optimistic, pointing out that no less than 21 bills have been filed to address the issue: “I think Virginia is going to change some things this year.”

Several legislators proposed bills in response to the Post series, which found of the 60 fatalities at day-care operations in the past decade, 43 died in unregulated settings. Many of the deaths involved risky actions by caregivers, The Post found.

In Virginia, people may legally run in-home day-care operations free of rules or standards if they watch no more than five children who are not related to them. Homes of caregivers licensed by the Virginia Department of Social Services, however, must have fire and safety inspections. They also are required to have working smoke detectors, a fire extinguisher and an evacuation plan.

Virginia ranks among the bottom eight states for its regulation of in-home day-care operations, according to Child Care Aware of America, a national child resource group.

The last significant legislative push to bring oversight to Virginia’s unregulated day-care operations was in the 1990s.

Across the state, there is a shortage of licensed day-care providers. Unregulated homes often provide cheaper, more flexible options for parents.

Because unregulated day-care operations are not tracked by state officials, the exact number of children in such care in Virginia is unknown. Child-care experts think that most children are in regulated settings.