BURBANK, Calif. — Irma Mangayan was lathering and rinsing a 92-year-old woman in Room 413 one recent afternoon when she received a page from another room. An incontinent resident had an accident, and Ms. Mangayan would have to clean it up.

Before her shift was over at Belmont Village Senior Living, Ms. Mangayan would hoist women and men into their wheelchairs, escort residents using walkers downstairs to the dining room and then back and perform myriad other tasks that they once could do for themselves.

Ms. Mangayan is a personal care aide, a grueling and low-paid profession that happens to be one of the country’s fastest growing. It is also increasingly filled with foreign-born, low-skilled workers like Ms. Mangayan, the kind now at the center of a national debate on immigration.

A proposal favored by a number of Congressional Republicans and the Trump administration would replace the current family-preference immigration system, which critics call “chain migration,” with one that favors skilled immigrants, while reducing admissions over all. Democrats have balked at the plan, while some Republicans have insisted it be a condition of any bill that legalizes the undocumented young adults, known as Dreamers, who could soon lose their protection against deportation. Several attempts to reach a deal have failed this year, and a spending bill passed on Friday did not resolve the issue, leaving the fight over immigration reform for another day.