The most common and most likely answer to this crossword clue is the 8 letter word STROLLER
I've seen this crossword clue in The Daily Telegraph
S
T
R
O
L
L
E
R
More clues leading to the result STROLLER
- Itinerant player’s buggy?
- One sauntering along street’s given wave
- He walks with a baby carriage
- Holy person with top-class car, one in no hurry
- Person walking slowly along street with posh car following
- Child’s vehicle holy man placed in front of car
- Buggy in street ahead of luxurious car
- Rambler rosemarine’s going to train
- Posh car following street buggy
- Square 50s retro style about buggy
Buddy explains!

The straight part
We know that "Person walking slowly" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case person walking slowly = stroller.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 STROLLER mean?
STROLLER noun- A seat or chair on wheels, pushed by somebody walking behind it, typically used for transporting babies and young children.
- One who strolls.
- A vagrant.
- Men's semiformal daytime dress comprising a grey or black single- or double-breasted coat, grey striped or checked formal trousers, a grey or silver necktie, and a grey, black or buff waistcoat.