the mistake is that you give the files to the grep command and not the tail.
the tail -f needs to get the files as input. try:
tail -f /var/links/proc2/id/myprocess*/Daily/myprocess*.log | grep --line-buffered "Search this: "
to get also the file names (however it will not be like grep output it is):
tail /var/links/proc2/id/myprocess*/Daily/myprocess*.log | grep --line-buffered -e'^==> .* <==$' -e'Search this: '