The main textual sources for the first ten memorials are The Scripture on the Ten Kings 十王經 and The Scripture on the Causes of Kṣitigarbha Bodhisattva Giving Rise to the Thought of Enlightenment and the Ten Kings 地藏菩薩發心因緣十王經. According to these texts, at each of the first ten junctures the spirit of the deceased enters the court of one of the Ten Kings 十王 for judgement. In Japan, the Ten Kings were each paired with a Buddhist deity according to the logic of 'original ground, manifest traces' (honji suijaku) 本地垂迹 in which each of the Ten Kings was envisaged as the “manifest trace” of Buddhist deities that were their “original ground.” At the last three junctures, the spirit of the deceased comes before three Buddhist deities.

The first seven memorials are well documented in Japan during the Nara (710–794) and Heian (794–1185) periods. The addition of the next three memorials (the one hundredth day, first and third year) began in late twelfth century, leading to the establishment of the Ten Memorials 十佛事. The addition of the last three memorials (seven, thirteen, and thirty-three years) began in the thirteenth century, leading to the establishment of the Thirteen Memorials 十三佛事.

References:

Tokuno Takayuki 徳野崇行. 2018. Nihon Zenshū ni okeru tsuizen kuyō no tenkai 日本禅宗における追善供養の展開 . Tokyo:  Kokusho Kankōkai.