Someone can't tell you where they live or how they got there. What's actually happening, and what gets you to a name.
Notice the pull to resolve this fast. It reads as a low-priority call eating your shift, and the quickest visible path is a transport or a handoff. The quickest real path is usually four minutes spent getting a name, which is slower at the start and much faster at the end.
Sudden confusion over hours, one-sided weakness, facial droop, slurred speech alongside those signs, a head strike, or low blood sugar point to a medical emergency, and none of what follows applies until that's been excluded. Exposure matters too. Someone who has been outdoors for an unknown stretch in heat or cold may be further along than they look, and hypothermia and dehydration are among the leading causes of death in this population when they go missing.
Past that, the useful thing to understand is that "lost" is your frame, not theirs. Many people with dementia found wandering don't experience themselves as lost at all. They're on a task, and the task is real even when its facts expired decades ago: getting to work, picking up children now in their fifties, going home to an address sold long ago. This matters operationally. Someone who believes they're lost accepts help. Someone who believes they're on their way somewhere experiences you as an interruption.
One more thing worth knowing before you decide what you're looking at: people with dementia who are lost are frequently mistaken by both the public and by responders for someone who is homeless, intoxicated, or simply out on an ordinary errand. Officers in one study described exactly this, including cases where days passed before anyone recognized that the person was missing at all.
Unknown time outdoors in heat or cold, any sign of dehydration or exposure, an injury they can't account for, or confusion worsening while you're with them all mean a medical evaluation rather than a ride home. If nobody can be identified and there's no safe handoff, that belongs with your agency's protocol, not this page.
Nobody leaves the house intending to be found confused in a parking lot. Whatever they were doing made sense when they started, and some version of it still does to them.