Are there any benefits to relieving oneself over a period of days, such as "three days or more," mentioned in authentic hadiths?

The question is unclear, but if what is meant is the existence of specific dhikr (remembrances) or du'a (supplications) for fulfilling a need within a certain period, then no such thing has been revealed. One who desires to fulfill a need should take the means for du'a to be answered, perform the Salat al-Hajah (Prayer of Need), and persist in supplication. Their du'a will be answered in one of three ways: the need is expedited, or it is stored for the Hereafter, or an equivalent evil is averted from them, provided they do not supplicate for a sin or the severance of kinship ties.

If what is meant are the benefits of expediting the fulfillment of Muslims' needs, then this is among the best acts of righteousness, and its benefits are great, including that Allah will assist one who strives to assist their brother. However, there is no specific period designated for fulfilling needs, nor is there any evidence for a specific timeframe in which the results of such actions appear. Allah alone knows when and how He rewards.