The tragical procession

[L Verge]

There are so many descriptions of a "simple pine coffin" transporting the body to the White House, that I think we can go with that. I have also seen references to everything from a caisson to a hearse to a farm wagon being the means of transportation. I haven't been to the dictionary, but could a bier mean anything that is holding a coffin? Finally, I found this about a former exhibit at the Chicago Historical Society that really begs the question of whether a flag covered the coffin or the body:

Charles F. Gunther Collection, 1920.1689
This U.S. flag, on view in Abraham Lincoln Transformed, has a unique history. Union army soldiers, in a gesture of deep sorrow and great respect, reportedly wrapped Lincoln’s body in this flag before placing it in a temporary pine coffin for transport from the Petersen House to the White House. The flag’s history was revealed by piecing together a number of sources.

It is part of the Charles F. Gunther Collection purchased by the Chicago Historical Society in 1920. Gunther, an avid collector of Americana, obtained the flag in 1892 from Mrs. Sarah L. Brearley, widow of William Brearley, Chief Clerk of the Military Store Keeper’s Department during the Civil War.
In a sworn affidavit, dated December 30, 1890, Mr. Brearley stated that the flag “was used to cover President Lincoln’s body while being carried (from the Petersen House, where Lincoln died)…to the White House,” and that it “came into my possession in my official capacity.”
Brearley identified the flag as a U.S. Army recruitment flag. It never flew over the Petersen House but was brought there, perhaps by the Union army soldiers summoned by Secretary of War Edwin M. Stanton during Lincoln’s death vigil.
The April 17, 1865 edition of the Baltimore Clipper reported that the “president’s body was removed from the private residence opposite Ford’s Theatre…at half-past nine o’clock…wrapped in the American flag.”
Elizabeth Dixon, wife of U.S. Senator James Dixon of Connecticut, who was with Mary Todd Lincoln at the Petersen House when Lincoln died, and later at White House when his body arrived, stated in a letter written May 1, 1865, that “…the remains of the murdered President…were taken into the great State bedroom, wrapped in an American flag.”
The flag has 34 stars that represent the number of states in the Union from 1861 to 1863. Early in the war, some people urged Lincoln to remove eleven stars for the seceded Southern states. Lincoln refused, in the belief that secession was not valid.
The flag has not been on public view in more than fifty years; it required a great deal of conservation work in order to be safely displayed. After Abraham Lincoln Transformed closes, the flag will be taken off its mount and returned to storage. This exhibition offers a rare opportunity to not only see this special flag, but many other important artifacts related to America’s 16th president.

blog.chicagohistory,org/index.php/2009/10/a-special-flag/