Promoters say the cards do encourage saving. Greenlight lets parents automatically match money the child sets aside in a savings account. Rates on bank savings accounts are still low — 2 percent is considered generous — making compound interest a hard sell. But parents can direct the app to pay the child their own, personal interest rates — say, 20 percent or even 100 percent — as an incentive. (The card account itself doesn’t pay any interest.)

Greenlight also allows parents to supervise spending by choosing the type of stores or restaurants where children can shop. Money on the card is put into two categories: for spending anywhere, or for only preapproved stores and websites. If the child tries to buy something at a nonapproved store — or tries to spend more money than is available on the card — the purchase is declined.

Children with mobile phones can get their own version of the app, which lets them check balances or seek a parent’s permission to buy a specific item.

Greenlight, with financial backers that include Amazon and two big banks, became available in 2017 and now has about 200,000 paying customers, said Tim Sheehan, the company’s chief executive. Next week, the card will add new features, including the ability to use the card at A.T.M.s.

Current, backed by investors including an arm of Fifth Third Bank, lets parents offer their children the option to earn money by doing chores. One example: “Mow Lawn. $10.” Once the chore is done, funds are transferred. Stuart Sopp, Current’s founder and chief executive, said his 9-year-old daughter had a card and often told him, “I want more chores.”

“It’s a parental toolbox,” he said.

The cards can offer a learning experience as long as parents don’t go into “helicopter” mode and overdo their control, said Bill Dwight, founder of FamZoo, a longtime family finance and budgeting program that added a debit card to its menu several years ago.

The idea is to let children make spending mistakes, with guardrails to prevent disasters, Mr. Dwight said. “If you nag the kid about every transaction,” he said, the child may tune you out.