tail multiple files and grep the output

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: '

You should have a look at multitail tool (Install using sudo apt-get install multitail)

In short, with multitail, you need to use the --mergeall flag for viewing output of all in one place

multitail --mergeall /var/links/proc2/id/myprocess*/Daily/myprocess*.log  | grep --line-buffered "Search this: " 

You can do the same without using grep

multitail -E "Search this: " --mergeall /var/links/proc2/id/myprocess*/Daily/myprocess*.log  

To view the output individually using multitail, this will give the filename as well.

multitail -E "Search this: " /var/links/proc2/id/myprocess*/Daily/myprocess*.log