I put on eight yards of warp to weave rag placemats. As I wove, each placemat measured 20 inches. When all eight mats were taken off the loom, the ends were stitched and the entire piece was washed in the washing machine. Then the piece was dried, ironed, and measured. I noticed then that the amount of shrinkage varied along the length of the long piece—with the beginning placemats shrinking much less than the later ones—there was as much as an inch difference between the first and last ones. Does this have something to do with the finished piece getting thicker on the cloth beam as the warp progressed?
I can't think of any way that the diameter of the cloth on the cloth beam would affect shrinkage. Also, the differences in the final placemat measurements are not likely to be related to actual shrinkage (at least not in terms of the materials; they would shrink at a consistent rate).
My guess is that you changed your beat as you went along. If the picks per inch are different, the finishing process affects the piece differently. If, for example, there are fewer picks per inch, there is more room for the warp yarns to move (both from shrinkage and from relaxing warp tension). To be sure of this, however, you'd have had to measure the picks per inch as you wove and then observed whether relaxing warp tension caused some of the difference. It might be interesting (and a clue!), however, to count the total number of picks in the first and last placemats. If they have the same number, we'll have to rethink this possible explanation! Weaving detectives!