The time-slice would by definition fit best with the running mean that uses the exact same time period. I must have misunderstood your question. Please try to explain again.
Do you move your time slices back or forth in time to check, wether results change?
We found e.g. for extremes in wind speed, that moving a time slice from e.g. 2040-2069 to 2041-2070 can change the results considerably
I see, when it comes to extremes, there can be substantial sensitivity to the time period. We have not explored this, but we are ingesting daily resolution river discharge, as well as bias adjusted temperature and precipitation into CDS (TBA). Thanks for raising this, and it should be included in the guidance for interpreting information on changes in extremes.