You can't move a file, AND edit it at the same time, since moving a file doesn't physically move the data (on the same filesystem), it just moves a pointer to the data. You can copy and convert the data, then delete the original file, or you can edit the original file, then move it.
cd SOURCE
for i in *.csv
do
awk 'NR>2{print s} {s=$0}' < "$i" > ../DESTINATION/"${i}"
rm "${i}"
done
If you omit the rm line, it gives you the opportunity to verify that everything was converted the way you want, before you delete the source files.