When looking at India it's Hinduism and then Islam, but no Buddhism despite the fact that it's founder was born there. Why is this?
There were kingdoms and eras in India where Buddhism was a major state religion. This is part of how Buddhism spread into Central, Southeast, and East Asia, via missions sent by powers like the Kusan Empire and trade routes. There were temple/monastic complexes in northern India that were the size of cities, and some Buddhist presence in southern India probably endured later than elsewhere. So first, it's not entirely correct to say that there was never a 'Buddhist era' in India- it's just that Buddhism almost always was co-existing with other, non-Buddhist, traditions that waxed and wained in royal favor and popular support over several centuries. We also have very incomplete records from this era of history, so it is difficult to identify with certainty how certain rulers in India and it's hinterlands projected themselves and were perceived- certainly Buddhist motifs figure in the official symbols of a number of rulers from Northwest India and other significant kingdoms, but a king might depict themselves and govern their kingdom using a combination of Buddhist, Central Asian, Greek, and non-Buddhist Indian symbols and deities.
Two factors are usually sited in the decline of Buddhism in mainland India:
During late antiquity and the middle ages, India saw the development of a couple of major religious movements- bhakti and tantra- that took hold in the popular imagination and introduced changes into how religion was practiced. Buddhism participated in the tantra movement, but this may have contributed to it being confused with and assimilated to mainstream Hindu practice. The bhakti movement seems to have re-invigorated the popular worship of gods, which may have declined compared to reverence for monks and holy people during the religious ferment of the Buddha's era, when there were several competing non-Brahmanist, non-vedic religious traditions that seem to have attracted attention and support.
With the destruction of the major university and monastic complexes in the north, the deaths of many senior monks and the scattering of those that remained into Tibet and Central Asia, there was no longer a central source of influence and power in the mainland subcontinent. Buddhism persisted in the maritime South and Sri Lanka, but suffered numerous reversals of fortune in the meantime- organized Buddhism was nearly wiped out in Sri Lanka on a couple of occasions, and monks had to be re-introduced from Southeast Asian lineages that had been established earlier, and may have been kept alive in Sri Lanka because it was a way for the Sinhala rulers there to distinguish themselves and prevent themselves from coming too much under the influence of Hindu kingdoms in Southern India.