This question is very broad.
Buddhism came to SE Asia after Hinduism therefore I think a better question is how did Buddhism integrate into SE Asian cultures rather than how did SE Asian cultures integrate into Buddhism.
For example, in Thailand, although there are thousands of Buddhist monks, the Buddhist monks do not aggressively teach Buddhism to the common people therefore common people still have many traditional beliefs based in animism, ancestor worship and Hinduism.
Before Hinduism and Buddhism were introduced into Thailand there was only one religious belief, that belief was that a spirit world
existed, not only did the spirits exist but they were mightily
powerful and controlling, this belief is called Animism and it
manifests itself in the form of spirit worship. It might not be
strictly correct to call Animism a religion, maybe it is better
categorized as a spiritual belief (spirit worship), but it is without
doubt the oldest form of worship known to mankind, ‘Spirit Worship’ in
one form or another was practiced long before all the popular world
religions.
The majority religion in Thailand is Buddhism, 95 percent of the population are Buddhist but the percentage of Thais who have animist
beliefs is probably slightly higher. Animism is practiced on a daily
basis by most Thai people, although their religion is Buddhism they
actually devote more of their time to Animist beliefs than they do
practicing Buddhism.
Buddhism is not the official or national religion of Thailand.
For example, the kings of Thailand are considered to be the reincarnations of the Hindu god Vishnu and many Thai national emblems are Hindu.
In Bangkok is the well-known Erawan Shrine of Brahma.
In my experience in Thailand, monks mostly teach the common person to be generous, to not be angry & to have metta (loving-kindness & tolerance). Monks generally do not even teach morality, let alone higher Buddhist teachings, unless these are requested by the people.
Of all SE Asian countries, in my experience, the most authentic Buddhism is found in Thailand, possibly because the Thai have always been a very independent people; they can act objectively.
Therefore, one major influence of Thailand on Buddhism is Thai teachers, most notably Buddhadasa Bhikkhu & also Ajahn Chah (in terms of doctrine) and Ajahn Mun (in terms of monastic asceticism), was returning to the original essence of Buddhism.
For example, the Supreme Buddhist Patriarch of Thailand once said the common interpretations of heaven & hell are cultural rather than Buddhist, below:
On this very subject, His Holiness
critically analyses that the very concept and belief of heaven and hell in Buddhism is a cultural
influence of indigenous culture and belief. He states: (I quote) ‘the subject of cosmology appeared in
Buddhism is clearly can be seen that it is not ‘Buddhist teaching’ at all but an ancient geography. The
concept and belief about it was included in Buddhist Canon merely because of strong influence of
popular belief of the time. Later Commentaries further explain about heaven and hell in a greater
detail distant itself from the original teaching of the Buddha. If Buddhism teaches such belief on
heaven and hell it would not be Buddhism at all but an ancient geography. Buddha wouldn’t be the
Buddha who delivered the Noble Truth and ‘timeless’ message for mankind.’ (p. 1) (end of the
quote) He then shows in his teaching that the concept of heaven and hell in Buddhism are in fact
symbolic, representing the quality of mind and spirituality instead. One can be in heaven and hell in
this very earth and life. No need to wait until one dies. 28 September 2008
By Ven. Phra Anil Dhammasakiyo