The most common and most likely answer to this crossword clue is the 5 letter word LYSIS
I've seen this crossword clue in The Guardian
L
Y
S
I
S
More clues leading to the result LYSIS
Buddy explains!

The straight part
We know that "destroying process" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case destroying process = lysis.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 LYSIS mean?
LYSIS noun- A gradual recovery from disease (opposed to crisis).
- The disintegration or destruction of cells
- The breakdown of molecules into constituent molecules
- A plinth or step above the cornice of the podium in an ancient temple.