The Rama (יו”ד קכא:ב) writes that the custom is to do hagala only when a utensil is not a ben yomo – not used in the past twenty-four hours. There are two reasons for this custom. One reason is that we don’t want a person to mistakenly do hagala on a meat utensil and dairy utensil at the same time. If someone mistakenly did hagala on both at once, the beliot of the meat and dairy would go into the water and become a forbidden mixture of meat and milk. The water would then have a forbidden status, and in turn it would make the utensils being koshered absorb beliot of the forbidden water. If the utensils being koshered are not ben yomo, even if they went into the pot at once there would not be an issue of meat and milk mixing, because the beliot in the pots are already weakened and can no longer become forbidden. [For a more comprehensive explanation see Ben Yomo]
A second reason is that if a person does hagala to a utensil with beliot that are forbidden, and there is not sixty times more water than the volume of the utensil being koshered, the forbidden beliot will go through the water of hagala and forbidden beliot will go into the pot being used for hagala. The pot will subsequently put forbidden beliot into any utensil placed inside the pot. Here too, if the pot being koshered is not ben yomo, the beliot in it will become distasteful and will no longer be forbidden. Therefore, even if there is less water being used for the hagala than sixty times the volume of the utensil being koshered, the beliot will not make the water, and pot used for the hagala, forbidden.
If one accidentally did hagala on a ben yomo utensil, and there was sixty times more water than the utensil being koshered, the hagala works (שו”ע או”ח תנא:ב). If one is doing hagala for Pesach, the same halachot apply just like the rest of the year. However, when koshering on Pesach itself, hagala does not work, even if the pot was not a ben yomo and there was sixty times more water than the volume of the utensil being koshered. This is because chametz on Pesach is forbidden, even when it is a minute amount in a large mixture, and it is forbidden even if it is distasteful.
Because we are always careful to do hagala on utensils that are not ben yomo, one does not need to have sixty times more water than the utensils being koshered.
The twenty-four hours of ben yomo in regards to chametz (and all permitted foods, such as dairy or meat) start from the last instance one used the utensil for the chametz, regardless of the utensil having been used after that for boiling water or other non-chametz foods. If one is koshering a utensil that had forbidden beliot in it, one needs to wait twenty-four hours from the last time the utensil was used for any hot food, even water. This is because when the previously forbidden beliot in the pot touch the water, they give the water the same forbidden status. The now forbidden water puts fresh beliot back into the pot, thus restarting the clock on the twenty-four hours.
For Sephardim, forbidden beliot can only give another food the same forbidden status if the beliot come from a forbidden mixture of milk and meat. Any other beliot, from any other forbidden food, do not do this. Therefore, the clock does not restart on ben yomo when cooking other permitted foods in it, unless the original forbidden beliot came from a mixture of meat and milk.