What it means
To mosey is to move in a slow, easy, no-stress way, usually because you're in no rush and not trying to power-walk your life away. If someone says mosey on over, they're telling you to come by nice and easy. It can also mean drift along or move on without much urgency. Very laid-back, very Southern, very unbothered.
Usage examples
"I told Bubba I’d be there at six, but I was still moseyin’ through H-E-B at 6:20, debating brisket like time was optional."
"We had loads of time, so we just moseyed along the river and stopped for ice cream twice."
"Mosey on over when you're ready, no rush, the kettle's on and the dog wants a fuss."
"We ain't gotta be there yet, so let's just mosey down Main Street and see what's still open."
"He moseyed into the cookout an hour late, plate ready, acting like the afternoon was gonna wait for him anyway."
Where it comes from
Mosey shows up in American English in the early 19th century, with the sense of moving along in a slow, relaxed way. It’s long been tied to Southern and Western speech, where it kept that easygoing, no-hurry feel and settled in as a classic word for ambling about.