The most common and most likely answer to this crossword clue is the 8 letter word TELEPORT
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More clues leading to the result TELEPORT
Buddy explains!

The straight part
We know that "Move thoughtfully" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case move thoughtfully = teleport.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 TELEPORT mean?
TELEPORT noun- A teleporter.
- A satellite ground station.
- To travel, often instantaneously, from one point to another without physically crossing the distance between the two points.
- To move (an object) in this fashion, as by telekinesis.