Grep command to find files containing text string and move them

Use xargs in concert with mv's third syntax: mv [OPTION]... -t DIRECTORY SOURCE...

grep -lir 'string' ~/directory/* | xargs mv -t DEST

Be careful about files containing special characters (spaces, quotes). If this is your case, filtering the list with sed (adding quotes around filenames with s/^/'/;s/$/'/) might help, but you'd have to be sure, these quotes won't appear in the filenames. GNU grep has the -Z/--null option to NUL-terminate filenames.

An alternative to the third syntax for mv is using xargs with the placeholder string (-I).

Another option is command substitution - $( ) or backticks `` (in bash) as mentioned in ire_and_curses' answer.

As always, beware of grep -r. -r is not a standard option, and in some implementations like all but very recent versions of GNU grep, it follows symbolic links when descending the directory tree, which is generally not what you want and can have severe implications if for instance there's a symlink to "/" somewhere in the directory tree.

In the Unix philosophy, you use a command to search directories for files, and another one to look at its content.

Using GNU tools, I'd do:

xargs -r0 --arg-file <(find . -type f -exec grep -lZi string {} +
  ) mv -i --target-directory /dest/dir

But even then, beware of race conditions and possible security issues if you run it as one user on a directory writeable by some other user.

If your file names don't contain any special characters (whitespace or \[*?), use command substitution:

mv `grep -lir 'string' ~/directory/*` destination/