These urns were quite independent from the crematory aspect as they rarely contained ashes but were used as symbols only.
An urn is reverential, sombre, elegant and has class. Grecian urns made it into many rich houses in Scotland in the 19th century. The idea behind it was created by the great romantic poets of the English tradition and their love of the past. The urn becomes the “cold pastoral”.
John Keats’ Ode on a Grecian Urn
Thou, silent form! dost tease us out of thought
As doth eternity: Cold Pastoral!
When old age shall this generation waste,
Thou shalt remain, in midst of other woe
Than ours, a friend to man, to whom thou say’st,
‘Beauty is truth, truth beauty, – that is all
Ye know on earth, and all ye need to know.’(Arthur Quiller-Couch, ed. 1919. The Oxford Book of English Verse: 1250–1900)
Often the urns are draped with a veil, cloth or shroud. Works on mortuary art differ considerably in their views on what the veiled urn actually means.
“Often, the graveyard urn is draped with a veil either symbolising the veil between earth and heaven or used as a reverential accessory.”
(Douglas Keister: Stories in Stone; Gibbs Smith, 2004, p.137).
“The cloth draping the urn symbolically guarded the ashes. The shroud-draped urn is believed by some to mean that the soul has departed the shrouded body for its trip to heaven. Others say that the drape signifies the last partition between life and death.”
I would argue from the point of style, which was in itself one of the reasons of choosing that particular design and the shroud or veil then giving the urn an even more classy, slighty Romanesque touch. Since urns (veiled or plain) often make the highest points of a graveyard, I would also argue the case that it might simply be a symbol of power and influence of the family erecting the headstone. They chose an urn that rises elegantly towards heaven.
An urn is the epitome of elegance, class and style and had soon replaced the skull and crossbones on many graveyards in Scotland and the world. That is especially true for large cemeteries in cities like Glasgow, Edinburgh or Inverness, where the rich and famous were interred. But it is just as much true for the rural parts of the North of Scotland.
James Deetz and Edwin S. Dethlefsen claim (after having examined 25 000 headstones in New England), that the urn and willow became popular in Boston right at the beginning of th 19th century, replacing the Cherub and Death’s Head. The changed apparently occurred in harmony with those in England. Scotland is not mentioned. The change has its reasons ”in the ecclesiastical history of New England. The period of decline of death’s head’s coincides with the decline of orthodox Puritanism.”
The Plymouth Colony Archive Project: Death’s Head, Cherub, Urn and Willow, www.histarch.illinois.edu/plymouth/deathshead.html
The urn was in its origin a means of storing the ashes in case of cremation. The container then developed into a stylish symbol of death, many would choose in the 19th century. It also has a certain safety aspect, carefully enclosed in an urn like a body would be in a coffin, without rot and mould, just pure and clean stone or marble. The roundness bringing a softer aspect into the harsh reality of he business.
In Scottish rural areas on the other hand, many headstones sprout a variation of the classical rounded urn, square with edges, much harsher. Too fancy would not do in those days.