The air gets colder as you get higher: Arnison Crag

We went into the hills for the first time this year, climbing small hills above Patterdale. No temperature inversion here: the air clearly gets colder as you get higher hence the precipitation is snow higher up. You can see a few flakes in the second photograph. So does the precipitation start as snow and then melt on the way down to lower levels? Is it just because it has to fall further and down into the warmer air? And why would the air be warmer closer to the ground on a miserable day like today with no sun?