The Gelug school holds firmly to the gradual side of the question settled at Samye, that awakening is reached by staged analysis and cultivation and not by a single non-conceptual leap.
Source Indo-Tibetan Buddhism: Indian Buddhists and Their Tibetan SuccessorsDavid L. Snellgrove
Kagyu, Denominations Why this is disputed. Whether the Kagyu presentation of mahamudra is a sudden path was attacked from inside Tibet and has never been settled. Sakya Pandita's thirteenth century critique charged that Kagyu mahamudra instruction resembled the Chinese sudden enlightenment teaching condemned at the Council of Lhasa, a charge Kagyu teachers have answered without the dispute ever being laid to rest.
Source Introduction to Tibetan BuddhismJohn Powers
Dzogchen is presented as a path of direct recognition rather than of staged construction, which places the Nyingma on the sudden side of the question Samye framed, though the school's own literature resists that framing.
Source Indo-Tibetan Buddhism: Indian Buddhists and Their Tibetan SuccessorsDavid L. Snellgrove
The Rinzai line descends from the Linji house of Chinese Chan and its koan curriculum is built around a breakthrough rather than an accumulation, which is the sense in which it is called the sudden school.
Source Zen Buddhism: A History, Volume 2: JapanHeinrich Dumoulin
Sakya Pandita's attack on Gampopa's mahamudra as a Chinese-style sudden path is the school's defining intervention on this question and puts the Sakya firmly on the gradual side.
Source Introduction to Tibetan BuddhismJohn Powers
Soto, Denominations Why this is disputed. Whether Soto is the gradual school is argued, and the textbook answer does not survive the texts. Dogen's own writing describes practice and realization as one and the same act rather than a gradual approach to a later goal, which sits uneasily with the standard Soto-gradual, Rinzai-sudden pairing still often repeated outside specialist scholarship.
Source Zen Buddhism: A History, Volume 2: JapanHeinrich Dumoulin
The Theravada path is gradual in cultivation but its decisive moments are instantaneous: the path-consciousness at each of the four stages arises in a single mind-moment. The tradition therefore does not divide on this question in the way the East Asian schools do.
Source The Foundations of BuddhismRupert Gethin
Zen inherits the Chinese sudden teaching from the Southern school of Chan, though the sudden and gradual houses of that dispute are now read as a polemical construction of the eighth century rather than as two real lineages.
Source Seeing Through Zen: Encounter, Transformation, and Genealogy in Chinese Chan BuddhismJohn R. McRae