Do they do rituals and prayers for their loved ones?
Do they cremate their dead?
Oops. Forgot to answer this part.
I am unfamiliar with any modern customs, but ancient customs varied by region, period and religion. The part with which I am most familiar pertains to customs around Dali and Kunming in Yunnan.
In these parts of Yunnan, Buddhist grave yards of the Song through Ming dynasties have been found to contain both cremated remains and full burials. Cremated remains were sometimes found in a jar that was placed inside a larger jar. Occasionally, there were three of more of these nested vessels in a single grave. Jars that did not contain remains were also found around the body in some of the full burial graves.
Though many of the pottery jars lack any inscription, some that contained remains were painted or incised with prayers in Sanskrit (using what appears to be an early type of Devanagari script) as well as Chinese, and some even have writing identified as shamanistic inscriptions on the bottom. These same jars often have vajra and/or the animals of the Chinese zodiac on them. Pottery offering jars contained objects such as cowrie shell money, gold leaf, rice, blue glass, metal plaques of bronze, silver or gilded-bronze, and small bronze discs (sometimes identified as mirrors due to their highly polished surfaces). Some of these jars held only the offerings, while others contained offerings with the cremated remains.
Some of the excavated metal containers from the same grave complexes have been made from arsenical copper, and have contained dharmachakra painted on fabric or paper, as well as on the metal. Contents have included metal plaques with prayers in Chinese. There have also been several sacrificed turtles among the jars I have studied. All been small water turtles, and the heads have not been found.
On both pottery and metal containers, the lotus blossom was a major theme, either painted, incised or embossed. The same is true for the dharmachakra.
As for the prayers on the incised pottery vessels, the portions written in Chinese tended to be formulaic. Translation is difficult, since the text appears rather poetic, leaving much to be inferred. We have received translations so different that you would never suspect the people were looking at the same text. Two of the more common prayers are:
追為亡者 XX 神道
zhūi wéi wáng zhě X X shén dào (XX=name of the deceased)
Best translation thus far: Seeking enlightenment (the spirit path) on behalf of the deceased XX.
佛頂尊勝陁羅尼神咒
fó dĭng zūn shèng tuó luó ní shén zhòu
This one contains a transliteration of a Sanskrit mantra,
tuoluoni Best translation thus far: Buddha (says), in the highest respect, this most glorious holy incantation tuoluoni.
Though the Sanskrit writing on the jars runs in rings or a continuous spiral, the Chinese portion has been found written left to right, right to left, and top to bottom, and is sometimes broken into multiple lines. I have seen only one script vessel that was entirely written in Chinese characters, but they were used to write the Sanskrit language, which appears to have been more common in Yunnan at that time and place. On jars with both Chinese and Devanagari script, the Chinese is often far less neatly incised, and often displays variations that could be called mistakes. In contrast, the Devanagari script is always very neatly written.
Based on the scripts used on the artifacts, it is theorized that the region was strongly influenced by India during this time. As such, the burial customs likely have similar links.
Sorry I don't have information on any other region or time, but I hope this helps.
Melanie