Buddhist monks have been involved in funerary culture in East Asia for a very long time, in fact they were often involved in burials before Buddhism even spread to China. However, I'm really only familiar with it being a major stereotype in Japanese Buddhism, where Buddhism became a sort of funerary cult during the period of it's decline in the Edo period following a mild state repression.

When Buddhism entered Japan, it had to contest with the strong practice of Shinto, or rather proto-Shinto since Shinto was yet to be systematized or made into a recognizable form. During the 6th century, there was a serious effort by Prince Shotoku to make Buddhism the state religion and diffuse it widely. Thus, we saw an explosion of Buddhism in Japan around this period in history. However, this did not replace or even push out the native Kami-worship and in fact often had to combine or compete with it.

While Buddhism grew in power and prestige throughout the centuries, it became increasingly disconnected from the general populace, and when it underwent a sort of purge from government under the Tokugawa shogunate, who favored Noe-Confucianism as a state philosophy, Japanese Buddhism had to find a way to survive without the state patronage it had enjoyed and without the Sohei warlord-abbots who had controlled part of Japan during the turbulent eras before the relative peace of the Edo Period.

For a long time before this, Buddhism had gained a reputation as death cult for it's focus on death and rebirth. Whereas Shinto often encouraged the celebration of life and looked at death as something impure or wrong, Buddhism had no qualms about death and often took the practice of burial and funeral, which connected it to the populace. To the Japanese of the Edo and Sengoku periods, Buddhism was somewhat removed from daily life, outside of holidays and funerals. The corruption of Buddhism did not help matters and the often lax monks were more than happy with the disconnect. However, now Buddhism faced an issue of survival.

While it still had wealthy patrons, more and more it needed the patronage of the laity, now more focused on Shinto, to survive. Thus, they more heavily emphasized the death and funerary rituals that they had previously become known for and now were almost entirely known for funerals. This continued until the Meiji where Buddhism was truly repressed and stripped entirely from Shinto.

During the Meiji period, Shinto became the state religion and Buddhism was repressed and shunned by the government. Monks were disrobed en masse and several monasteries disbanded. This was the origin of Japan's "Buddhist priests", non-celebate, part time priests who cared for the temples and monasteries. This really is where Japan's contemporary association with funerary practices and part-time priests comes from in it's modern form.