Chasing record, leftist gradually came to the point (7)
I believe the answer is:
tapered
'gradually came to the point' is the definition.
(to taper is to gradually narrow or come to a point)
'chasing record leftist' is the wordplay.
'chasing' means one lot of letters go next to another (some letters chase after others).
'record' becomes 'tape' (synonyms).
'leftist' becomes 'red' (red can informally mean communist).
'tape' next to 'red' is 'TAPERED'.
(Other definitions for tapered that I've seen before include "Narrow" , "Narrowing to a point" , "Coming to a point" , "Narrowed by red tape" , "got slimmer" .)