“Ban the box” laws requiring employers to delay asking about applicants’ conviction history until far along in the job application process have gained momentum at the state and local levels and are leaving employers to face a variety of potential liability issues, say experts.

These range from fines that could be levied against them under the Fair Credit Reporting Act in connection with improper forms, to potential litigation surrounding the criteria used in determining which criminal convictions justify withdrawing job offers.

Twenty-nine states and more than 150 cities and counties have adopted these rules, say observers.

In their “purest” form, ban the box laws say “when you can ask on an application form, or during an application process or interview, about criminal conviction history,” said Stephen R. Woods, a shareholder with Ogletree, Deakin, Nash, Smoak & Stewart P.C. in Greenville, South Carolina.

Although the rules vary, “the latest point in the process” when the inquiry restrictions apply is when a conditional job offer has been made, he said. “Once you get to that stage,” with few exceptions, the inquiry restrictions no longer apply.

Experts say employers must keep on top of these regulations to avoid liability. These ordinances “all differ in their remedies, so depending on where the employer is located, they could be faced with a simple slap on the wrist or they could receive monetary penalties ranging from $100 to several thousand dollars,” said Jennifer Mora, senior counsel for Seyfarth Shaw L.L.P. in Los Angeles.

“And then several laws allow for private right of action,” which permits job applicants to file suit and obtain emotional distress and other compensatory damages and economic loss for employers’ failure to comply, Ms. Mora said. “They are definitely not laws that should be ignored.”

“Employers are getting hit in some of the riskier jurisdictions” including New York City, which has “been aggressive recently” on this issue, and the District of Columbia, she added.

The process “has become so overwhelming for the employer” in places such as New York that some “might be afraid to obtain a background check,” although they are exposing themselves to liability if they do not, said Christina A. Stoneburner, a partner with Fox Rothschild L.L.P. in Morristown, New Jersey.

New York’s Fair Chance Act, which took effect in October 2015 and is considered among the tougher laws, requires employers who deny an applicant a job because of a criminal record to explain why in writing and give the employee the opportunity to discuss the issue or correct any wrong information.

Ban the box rules create “a situation where an employer can walk into a risk by asking questions when they think they’ve done everything just right,” but in fact have violated local law, said David J. Woolf, a partner with Drinker Biddle & Reath L.L.P. in Philadelphia.

Another issue facing employers is the criteria they use to disqualify applicants who have a criminal conviction, say observers.

“That’s the area that’s getting more attention, and that’s the area that’s getting more litigation,” Mr. Woolf said.

Observers note that guidance from the U.S. Equal Employment Opportunity Commission says employers should use the criteria of whether a crime is “job-related and consistent with business necessity.”

Experts say no one would argue against an employer not hiring someone as an accountant who has been convicted of embezzlement. But many other cases are less clear-cut.

One example would be an applicant with an 8-year-old conviction of driving while intoxicated “and the employer says, ‘I don’t want anyone with any conviction working here. It shows disrespect for the law,’” Mr. Woolf said.

And what if the crime is a 20-year-old conviction for showing a fake ID while in college? While there are black-and-white issues, “there’s a lot of gray” as well in connection with this issue, he said.

“I have had a case where someone who had a conviction for selling drugs,” including its manufacture and distribution, “was looking for a job in a warehouse that supplied a lot of drugs,” he said.

“You would think” that the employer would be justified in not hiring that individual, “but a state agency ruled otherwise,” he said.

“The state took the view it was too old or he had been rehabilitated,” Mr. Woolf said.

While the employer ultimately prevailed on the issue in court, “it was a case that should never have gotten to first base,” said Mr. Woolf, who would not provide additional details.

“It’s good policy to be able to articulate why a specific criminal conviction prohibits someone from working in a specific position,” said Kimberly F. Seten, a partner with Constangy, Brooks, Smith & Prophete L.L.P. in Kansas City, Missouri.

Meanwhile, Indiana has approved legislation that bans its political subdivisions from enacting “ban the box” restrictions for private employers, noted Pamela Q. Devata, a partner with Seyfarth Shaw in Chicago.

Indiana Gov. Eric Holcomb has signed an executive order excluding criminal history questions from state employment job applications.

Ms. Devata said a bill comparable to Indiana’s has failed to pass the Texas Legislature, and so far no other states have followed Indiana’s example.