Is it possible to make find more efficient to use than Find + Grep in this scenario

You want to copy files of a specific type, to a specific directory? This is best done with xargs, and you don't even need that sh. This is a more appropriate way to do it, should also run more efficiently.

find "$sourcedir" -type f -name "*.type" | xargs cp -t targetdir

If you need to handle special file names, then use NULL as your seperator

find "$sourcedir" -type f -name "*.type" -print0 | xargs -0 cp -t "$targetdir"

I'm not sure:

grep -r -i 'the brown dog' /*

is really what you meant. That would mean grep recursively in all the non-hidden files and dirs in / (but still look inside hidden files and dirs inside those).

Assuming you meant:

grep -r -i 'the brown dog' /

A few things to note:

• Not all grep implementations support -r. And among those that do, the behaviours differ: some follow symlinks to directories when traversing the directory tree (which means you may end up looking several times in the same file or even run in infinite loops), some will not. Some will look inside device files (and it will take quite some time in /dev/zero for instance) or pipes or binary files..., some will not.
• It's efficient as grep starts looking inside files as soon as it discovers them. But while it looks in a file, it's no longer looking for more files to search in (which is probably just as well in most cases)

Your:

find / -type f -exec grep -i 'the brown dog' {} \;

(removed the -r which didn't make sense here) is terribly inefficient because you're running one grep per file. ; should only be used for commands that accept only one argument. Moreover here, because grep looks only in one file, it will not print the file name, so you won't know where the matches are.

You're not looking inside device files, pipes, symlinks..., you're not following symlinks, but you're still potentially looking inside things like /proc/mem.

find / -type f -exec grep -i 'the brown dog' {} +

would be a lot better because as few grep commands as possible would be run. You'd get the file name unless the last run has only one file. For that it's better to use:

find / -type f -exec grep -i 'the brown dog' /dev/null {} +

or with GNU grep:

find / -type f -exec grep -Hi 'the brown dog' {} +

Note that grep will not be started until find has found enough files for it to chew on, so there will be some initial delay. And find will not carry on searching for more files until the previous grep has returned. Allocating and passing the big file list has some (probably negligible) impact, so all in all it's probably going to be less efficient than a grep -r that doesn't follow symlink or look inside devices.

With GNU tools:

find / -type f -print0 | xargs -r0 grep -Hi 'the brown dog'

As above, as few grep instances as possible will be run, but find will carry on looking for more files while the first grep invocation is looking inside the first batch. That may or may not be an advantage though. For instance, with data stored on rotational hard drives, find and grep accessing data stored at different locations on the disk will slow down the disk throughput by causing the disk head to move constantly. In a RAID setup (where find and grep may access different disks) or on SSDs, that might make a positive difference.

In a RAID setup, running several concurrent grep invocations might also improve things. Still with GNU tools on RAID1 storage with 3 disks,

find / -type f -print0 | xargs -r0 -P2 grep -Hi 'the brown dog'

might increase the performance significantly. Note however that the second grep will only be started once enough files have been found to fill up the first grep command. You can add a -n option to xargs for that to happen sooner (and pass fewer files per grep invocation).

Also note that if you're redirecting xargs output to anything but a terminal device, then the grepss will start buffering their output which means that the output of those greps will probably be incorrectly interleaved. You'd have to use stdbuf -oL (where available like on GNU or FreeBSD) on them to work around that (you may still have problems with very long lines (typically >4KiB)) or have each write their output in a separate file and concatenate them all in the end.

Here, the string you're looking for is fixed (not a regexp) so using the -F option might make a difference (unlikely as grep implementations know how to optimise that already).

Another thing that could make a big difference is fixing the locale to C if you're in a multi-byte locale:

find / -type f -print0 | LC_ALL=C xargs -r0 -P2 grep -Hi 'the brown dog'

To avoid looking inside /proc, /sys..., use -xdev and specify the file systems you want to search in:

LC_ALL=C find / /home -xdev -type f -exec grep -i 'the brown dog' /dev/null {} +

Or prune the paths you want to exclude explicitly:

LC_ALL=C find / \( -path /dev -o -path /proc -o -path /sys \) -prune -o \
  -type f -exec grep -i 'the brown dog' /dev/null {} +