In my company we will not change the cost index with the exception of running short on fuel (less than the OFP says we should have) in which case we may tactically change it to a lower value to prevent and en-route landing.
Some airlines I know provide two OFP's to the crew one with a significantly higher cost index, the latter to be used in the event of a delay.
Lastly people like PACE labs are making EFB software that will advise en-route tactical changes of the cost index.
BA are the only airline I know of that changes their CI for t/o climb and cruise... I would be curious to the "briefing" that BA provide its pilots..l the theory of this idea