The most common and most likely answer to this crossword clue is the 4 letter word SPED
I've seen this crossword clue in The Sunday Telegraph
S
P
E
D
More clues leading to the result SPED
- Press person on latest news that’s travelled fast
- Ran exercises in South Dakota
- One’s pedometer shows how one went fast
- Rushed to spend without limitless money
- Ran over reckless pedestrians
- Hurried back to Oxford after supper occasionally
- Some reckless pedestrians belted along
- Dashed odds on journalist
- First signs of spelling problems and editor’s whipped
- Hurried outlay with no end of pain
Buddy explains!

The straight part
We know that "Quickly moved" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case quickly moved = sped.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 SPED mean?
SPED verb- To succeed; to prosper, be lucky.
- To help someone, to give them fortune; to aid or favour.
- To go fast.
- To exceed the speed limit.
- To increase the rate at which something occurs.
- To be under the influence of stimulant drugs, especially amphetamines.
- To be expedient.
- To hurry to destruction; to put an end to; to ruin.
- To wish success or good fortune to, in any undertaking, especially in setting out upon a journey.
- To cause to make haste; to dispatch with celerity; to drive at full speed; hence, to hasten; to hurry.
- To hasten to a conclusion; to expedite.