Here's a possible approach:
(*Find where all the "x" are. Since the replacement will depend on \
position, we can use these positions later.*)
Position[data, "x"]
(*You'll need some function that uses the position to determine a \
replacement value. This is just a placeholder for this example*)
ColumnBasedFn
(*Here's a function that uses the position to do a replacement on the \
data*)
TheReplacer[data_, position_] :=
ReplacePart[data, position -> ColumnBasedFn[Last@position]]
(*And finally, we apply the replacer with our positions. There are \
many ways to do this, I just happened to choose Fold.*)
Fold[TheReplacer, data, Position[data, "x"]]
This produces:
{{12, 27, 19, 44}, {11, ColumnBasedFn[2], 13, 43}, {ColumnBasedFn[1],
56, 23, 60}, {15, 37, 19, 52}, {10, 31, 21, ColumnBasedFn[4]}, {18,
40, 22, 43}, {15, 50, 28, 56}, {21, ColumnBasedFn[2],
ColumnBasedFn[3], 46}, {20, 39, 25, 51}, {ColumnBasedFn[1], 36, 23,
56}, {ColumnBasedFn[1], 27, 21, 39}, {14, 32, 20, 46}, {22, 40,
ColumnBasedFn[3], 49}, {ColumnBasedFn[1], 31, ColumnBasedFn[3],
44}, {ColumnBasedFn[1], 18, 21, ColumnBasedFn[4]}, {16, 23, 21,
34}, {27, 26, 23, 59}, {17, ColumnBasedFn[2], ColumnBasedFn[3],
35}, {13, 21, 20, 48}, {11, 51, 18, 54}}
