The most common and most likely answer to this crossword clue is the 4 plus 3 letter word DROP OFF
I've seen this crossword clue in The Guardian
D
R
O
P
O
F
F
Buddy explains!

The straight part
We know that "Fall asleep/ditch by side of road" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case fall asleep/ditch by side of road = drop off.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does DROP OFF mean?
DROP OFF noun- A sudden downward slope.
- A sudden decrease.
- A space reserved outside a bus or railway station for vehicles stopping to drop off passengers for onward transit.