Shareholder Death

Hi everyone,
Just wondering if someone can please clarify for me… what the ‘best practice’ is in the event of a shareholders death.
Should we be updating the shareholding from being beneficially held by the deceased individual to now being non-beneficially held in the name of the executor of the deceased estate (if known). Then once the probate of the will is provided, proceeding with share changes according to the will.
OR… should we be leaving the shareholding as beneficially held by the deceased individual, until such time as we recieve the probate of the will. Then at that time proceeding with share changes according to the will?