When independent directors are identified to fill a vacancy, relevant experience and how they could fit into the current board are key determinants. These days diversity is also a key criteria, but what happens when an independent director's reputation is brought into question due to outside or former activities? 

Former or current CEO/CFOs of FTSE 250 companies with relevant industry experience are highly sought after for the external outlook and knowledge  they can bring to a board. Indeed, in regulated sectors an individual who has not previously been vetted by a relevant body (like the FCA) is unlikely to be nominated to a board vacancy. 

Against experience comes reputation. At the time of appointment, discrete enquiries will be made but when a " clean" independent director is dragged into an unrelated governance issue, what should happen?

Most non executive appointment letters do not  give grounds for termination. Normally it is left to the chair or senior non executive to have a quiet word if considered appropriate but outside agencies have the right to force the pace. 

Activist shareholders often agitate for strategy change but rarely do they look at individual non executives' positions based on outside activities.

Should an independent director who had a role in a company previously, which has fallen on hard times, be automatically required to fall on his sword or should discretion be left with the board of which the individual is now part? They can best assess his contribution to the board.