The training by the man downstairs mostly is to do with race (6)
I believe the answer is:
ethnic
'is to do with race' is the definition.
(I've seen this before)
'the training by the man downstairs' is the wordplay.
'the training' indicates anagramming the letters (to train can mean to grow something into a shape).
'by' says to put letters next to each other.
'man' becomes 'N' (knight, a man in chess).
'downstairs' becomes 'ic' (I can't explain this - if you can you should believe this answer much more).
'the' put after 'n' is 'nthe'.
'nthe' anagrammed gives 'ethn'.
'ethn'+'ic'='ETHNIC'
'mostly' is the link.
This may not be right. It may be part of another bit of the clue.
Can you help me to learn more?
(Other definitions for ethnic that I've seen before include "from foreign cultures" , "To do with racial group or culture" , "from a particular community" , "Relating to race or culture" , "Relating to cultures" .)