Kapikulu means Slaves of the (Sultan’s) Gate, and Sipahis were the Ottoman cavalry. Kapikulu Sipahis were the state-maintained regular cavalry, as opposed to the Timarli Sipahis who were given fiefs and obligated to show up for campaigns. Although Janissaries (also Kapikulu) were recruited through the infamous Devshirme system, recruitment into the Kapikulu Sipahis seems to have been more voluntary, despite legally being slaves. From what I’ve sussed out, the first division (Sipahis) was recruited from the sons of Pashas and Beys, and the second division (Silahtars) was recruited from soldiers who performed great feats in battle. But that leaves the four other divisions, that is, the Right Ulufecis, the Left Ulufecis, the Right Garips, and the Left Garips. How were they recruited?
Great question!
The kapıkulu sipahis (also known as the altı bölük halkı, or "people of the six divisions") were recruited from two main sources during the sixteenth century: janissaries who merited promotion, and graduates from the imperial palace schools. While others were at times allowed to join the corps (as you note, some were the sons of grandees), these were where most of the soldiers came from. Thus, they were initially, like the janissaries, almost entirely devşirme recruits. While different divisions of the corps may have differed in which source of recruitment was most common, there doesn't appear to have been a strict distinction between them. Palace graduates, for instance, were recruited into any of the first four divisions (which constituted the vast bulk of its membership). The much smaller Right- and Left-Garib regiments are somewhat more mysterious. Garib means "stranger" and at least theoretically these were recruited from meritorious volunteers from the military frontiers, but as with the other divisions there does not seem to have been a strict distinction such that we can say definitively "X division contained men from Y source."
In the late sixteenth century the six divisions (along with the rest of the standing army) expanded dramatically for reasons historians are still debating, either socioeconomic changes internal to the empire or else the need for a larger army to fight the empire's multi-front wars. In any case, the corps expanded from 6,000 men in mid-century to 20,000 by the beginning of the seventeenth. As with the janissaries, the traditional recruitment methods couldn't keep up with this and admission to the corps of outsiders and the sons of existing cavalrymen became increasingly common, leading to the corps having a highly heterogeneous membership. As a consequence of its inclination toward factionalism and its burden to the treasury, the corps was politically suppressed and reformed by Köprülü Mehmed Pasha (grand vizier 1656-61) but due to its subsequent military and political insignificance in comparison to earlier in the century, it has not been seriously studied in this later period. The traditional methods of recruitment appear to have been given renewed emphasis, namely promotion from the janissary corps and graduates from the palaces. By this point, however, the devşirme had largely fallen out of use, and thus despite being recruited from the same sources, the kapıkulu sipahis of the late seventeenth century would have still been of mixed origin: Palace-educated foreign slaves, the sons of grandees, Muslim-born janissaries, the occasional devşirme recruit, etc.