A number of major factors contributed to the decline in Central/Southern Asia, I'm going to bullet them here:

• The rise of Hinduism

• Islamic conquests of Central/Southern Asia

• Lack of royal/governmental support.

Hinduism, which began to evolve from the old Vedic and Brahmanist traditions, heavily adopted parts of Buddhism. To this day, there's no clear distinction between Buddhists and Hindus in areas where they coexist - many Hindus worship Buddha as an avatar of Vishnu and many Buddhists worship the Hindu gods. During the Gupta Dynasty, Hinduism flourished heavily from the 4th-6th centuries CE.

At the time, early Buddhism was primarily geared towards monastics, and didn't have the same lay disciplines in South Asia that Hinduism possessed ( this also affected Jainism ). Brahmins, one of the major castes in India, wanted to absorb Buddhism into Hinduism to allow the dharma of both to meld and coexist - without unnecessary bloodshed.

Meanwhile, in Central Asia, the primary driving forces prior to Islamic conquests ( which we'll get to in a minute ) was the fall of the Tang Dynasty, which destabilized the regions.

Islamic expansion started in Northwestern India/Central Asia in the 7th century, with skirmishes between forces of the Kalif Umar and various Indian kingdoms and fiefdoms in modern-day Pakistan. These ended in mid-750 CE with the defeat of the Arabs and return west. While they were defeated in India, they conquered the Tarim basin, modern day Uzbekistan, Kazakhstan, Kygyzstan and Tajikistan. They plundered and burned the temples and killed those who refused to convert to Islam.

I should mention here that Islam has, in the Qu'ran, tolerance for non-Muslims, but only "People of the Book", i.e. Jews and Christians. Hinduism and Buddhism, with their iconography and unfamiliar ideals that lacked the monotheistic, revealed religion structure Muslims were used to, were treated harshly and barbarically - often people were forced at swordpoint. I do not have an anti-Islamic agenda in saying this - Islamic mosques do not depict Muhammed, Allah or any other living thing in their houses of worship because the Qu'ran and Sunnah do not allow it. This is why the Buddhists and Hindus were treated so harshly in this time period. It is necessary to tell history as it is so we do not repeat the mistakes of our past.

These resumed in the 11th century with Mahmud of Ghazni pushing into Peshwar, Pakistan. During this time Buddhist and Hindu temples were looted and people were forcibly converted to Islam. Islam historically had tolerance for Judaism and Christianity, "People of the Book" but not Hinduism and Buddhism which it saw as idolatrous due to rather common use of icons and elaborate statues/temples ( Buddhas of Bamiyan, in Afghanistan ).

While Central Asia itself was mostly conquered by the 8th century, they were not done in the immediate area. Next, Mu'izz al-Din, from Ghor, Afghanistan, in 1173 he began a campaign pushed into Punjab but was defeated in 1191. The following year he attempted again and succeeded, with his eyes on taking Delhi. He succeeded in establishing the Delhi Sultanate. He was killed in 1206, and the Mamluk Dynasty took the reins, from here Muhammad bin Bakhtiyar Khilji began attacking as far as modern-day Bangladesh, destroying Nalanda, a Buddhist university, burning and desecrating the temples and destroying years of documents. This was the fate of many other Hindu and Buddhist universities, which had coexisted for over a thousand years.

The Delhi Sultanate passed through a few dynasties, Mamluk, Khalji, Tughlaq, Sayyid, Lodhi to name the major important ones. The sultanate remained relatively stable, surving raids from the Mongol Empire, but losing Pakistan and Afghanistan to the Mongols. Tughlaq was the greatest extent, nearly reaching the tip of India, but it declined heavily after the Timurids sacked Delhi in 1398, before Timur Lang took it over finally in 1526, forming the Mughal Empire, which lasted until Britain took India from them.

After the Gupta Dynasty, there were not many Indian kings who were Buddhist, which set a poor example for the populace. People were unmotivated to follow Buddhism when their kings were Hindu.

Sources

The Sociology of Philosophies: A Global Theory of Intellectual Change

The Rise And Decline Of Buddhism In India

The Empire of the Steppes: A History of Central Asia.

The War of the Three Gods: Romans, Persians and the Rise of Islam

Medieval India: From Sultanat to the Mughals (1206-1526)

Tīmūr Lang, Encyclopaedia of Islam