In the case of the Dalai Lamas, they are each considered to be reincarnations of the bodhisattva Avalokitesvara (known as Chenrezig in Tibetan).
Other lineages of teachers / tulkus, such as Karmapas (who are also incarnations of Avalokitesvara), are the incarnations of different teachers or deities.
Je Tsongkhapa founded the first Gelug monastery, known as Ganden. In Sanskrit, this translates as “Tushita,” the pure land heaven that is ruled by the Buddha known as Maitre