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1 - Use syscolumns to dynamically generate our column lists for the unpivot operation; all values will be converted to varchar(max), w/ NULLs being converted to the string 'NULL' (this addresses issue with unpivot skipping NULL values)

declare @columns_max varchar(max),
        @columns_raw varchar(max),
        @cmd         varchar(max)

select  @columns_max = stuff((select ',isnull(convert(varchar(max),'+name+'),''NULL'') as '+name
                from    syscolumns
                where   id   = object_id('dbo.bigtable')
                and     name not in ('PK','UpdateDate')
                order by name
                for xml path(''))
            ,1,1,''),
        @columns_raw = stuff((select ','+name
                from    syscolumns
                where   id   = object_id('dbo.bigtable')
                and     name not in ('PK','UpdateDate')
                order by name
                for xml path(''))
            ,1,1,'')


select @cmd = '
insert #columns (PK, UpdateDate, ColName, ColValue)
select PK,UpdateDate,ColName,ColValue
from
(select PK,UpdateDate,'+@columns_max+' from bigtable
 union all
 select PK,UpdateDate,'+@columns_max+' from bigtable_archive
) p
unpivot
  (ColValue for ColName in ('+@columns_raw+')
) as unpvt
order by PK, ColName, UpdateDate'

--select @cmd

execute(@cmd)

--select * from #columns order by rid
;

select  c2.PK, c2.UpDateDate, c2.ColName, c1.ColValue as 'Old Value', c2.ColValue as 'New Value'
from    #columns c1,
        #columns c2
where   c2.rid                       = c1.rid + 1
and     c2.PK                        = c1.PK
and     c2.ColName                   = c1.ColName
and     isnull(c2.ColValue,'xxx')   != isnull(c1.ColValue,'xxx')
order by c2.PK, c2.UpdateDate, c2.ColName
;

1 - Use syscolumns and for xml to dynamically generate our column lists for the unpivot operation; all values will be converted to varchar(max), w/ NULLs being converted to the string 'NULL' (this addresses issue with unpivot skipping NULL values)

declare @columns_max varchar(max),
        @columns_raw varchar(max),
        @cmd         varchar(max)

select  @columns_max = stuff((select ',isnull(convert(varchar(max),'+name+'),''NULL'') as '+name
                from    syscolumns
                where   id   = object_id('dbo.bigtable')
                and     name not in ('PK','UpdateDate')
                order by name
                for xml path(''))
            ,1,1,''),
        @columns_raw = stuff((select ','+name
                from    syscolumns
                where   id   = object_id('dbo.bigtable')
                and     name not in ('PK','UpdateDate')
                order by name
                for xml path(''))
            ,1,1,'')


select @cmd = '
insert #columns (PK, UpdateDate, ColName, ColValue)
select PK,UpdateDate,ColName,ColValue
from
(select PK,UpdateDate,'+@columns_max+' from bigtable
 union all
 select PK,UpdateDate,'+@columns_max+' from bigtable_archive
) p
unpivot
  (ColValue for ColName in ('+@columns_raw+')
) as unpvt
order by PK, ColName, UpdateDate'

--select @cmd

execute(@cmd)

--select * from #columns order by rid
;

select  c2.PK, c2.UpdateDate, c2.ColName as ColumnName, c1.ColValue as 'Old Value', c2.ColValue as 'New Value'
from    #columns c1,
        #columns c2
where   c2.rid                       = c1.rid + 1
and     c2.PK                        = c1.PK
and     c2.ColName                   = c1.ColName
and     isnull(c2.ColValue,'xxx')   != isnull(c1.ColValue,'xxx')
order by c2.UpdateDate, c2.PK, c2.ColName
;

declare @columns_max varchar(max),
        @columns_raw varchar(max),
        @cmd         varchar(max)

select  @columns_max = stuff((select ',isnull(convert(varchar(max),'+name+'),''NULL'') as '+name
                from    syscolumns
                where   id   = object_id('dbo.bigtable')
                and     name not in ('PK','UpdateDate')
                order by name
                for xml path(''))
            ,1,1,''),
        @columns_raw = stuff((select ','+name
                from    syscolumns
                where   id   = object_id('dbo.bigtable')
                and     name not in ('PK','UpdateDate')
                order by name
                for xml path(''))
            ,1,1,'')


select @cmd = '
insert #columns (PK, UpdateDate, ColName, ColValue)
select PK,UpdateDate,ColName,ColValue
from
(select PK,UpdateDate,'+@columns_max+' from bigtable
 union all
 select PK,UpdateDate,'+@columns_max+' from bigtable_archive
) p
unpivot
  (ColValue for ColName in ('+@columns_raw+')
) as unpvt
order by PK, ColName, UpdateDate'

--select @cmd

execute(@cmd)

--select * from #columns order by rid
;

select  c2.PK, c2.UpDateDate, c2.ColName, c1.ColValue as 'Old Value', c2.ColValue as 'New Value'
from    #columns c1,
        #columns c2
where   c2.rid                       = c1.rid + 1
and     c2.PK                        = c1.PK
and     c2.ColName                   = c1.ColName
and     isnull(c2.ColValue,'xxx')   != isnull(c1.ColValue,'xxx')
order by c1.PK, c1.UpdateDate, c1.ColName
;

declare @columns_max varchar(max),
        @columns_raw varchar(max),
        @cmd         varchar(max)

select  @columns_max = stuff((select ',isnull(convert(varchar(max),'+name+'),''NULL'') as '+name
                from    syscolumns
                where   id   = object_id('dbo.bigtable')
                and     name not in ('PK','UpdateDate')
                order by name
                for xml path(''))
            ,1,1,''),
        @columns_raw = stuff((select ','+name
                from    syscolumns
                where   id   = object_id('dbo.bigtable')
                and     name not in ('PK','UpdateDate')
                order by name
                for xml path(''))
            ,1,1,'')


select @cmd = '
insert #columns (PK, UpdateDate, ColName, ColValue)
select PK,UpdateDate,ColName,ColValue
from
(select PK,UpdateDate,'+@columns_max+' from bigtable
 union all
 select PK,UpdateDate,'+@columns_max+' from bigtable_archive
) p
unpivot
  (ColValue for ColName in ('+@columns_raw+')
) as unpvt
order by PK, ColName, UpdateDate'

--select @cmd

execute(@cmd)

--select * from #columns order by rid
;

select  c2.PK, c2.UpDateDate, c2.ColName, c1.ColValue as 'Old Value', c2.ColValue as 'New Value'
from    #columns c1,
        #columns c2
where   c2.rid                       = c1.rid + 1
and     c2.PK                        = c1.PK
and     c2.ColName                   = c1.ColName
and     isnull(c2.ColValue,'xxx')   != isnull(c1.ColValue,'xxx')
order by c2.PK, c2.UpdateDate, c2.ColName
;

Note: apologies ... couldn't figure out an easy way to cut-n-paste the rextester output into a code block. I'm open to suggestions.