With GNU find(1), xargs(1), and grep(1):

• duplicate the directory structure:

    cd /path/to/dest
    find /path/to/src -type d ! -path /path/to/src -printf '%P\0' | xargs -0 mkdir -p
  

• restore permissions:

    rsync -a --existing /path/to/src /path/to/dest/
  

• copying files with the given pattern:

    cd /path/to/src
    grep -rlZ pattern | \
        (cd /path/to/dest; \
        xargs -0 sh -c ' \
            while [ $# -ne 0 ]; do \
                grep pattern "/path/to/src/$1" >"$1"; \
                shift; \
            done' sh)
  

With GNU find(1), xargs(1), and grep(1):

• duplicate the directory structure:

    src=/path/to/source
    dest=/other/path/to/destination
    pat='some_grep_pattern'
  
    cd "$dest"
    find "$src" -type d ! -path "$src" -printf '%P\0' | xargs -0 mkdir -p
  

• copy files with the given pattern:

    cd "$src"
    grep -rlZ "$pat" | \
        (cd "$dest"; \
        xargs -0 sh -c ' \
            while [ $# -ne 0 ]; do \
                grep "$pat" "$src/$1" >"$1"; \
                touch -r "$src/$1" >"$1"; \
                shift; \
            done' sh)
  

• restore permissions, assuming Linux, and assuming you don't have filenames with embedded newlines:

    cd "$src"
    getfacl -RPe . | (cd "$dest"; setfacl --restore=-)
  

With GNU find(1), xargs(1), and grep(1):

• duplicate the directory structure:

    cd /path/to/dest
    find /path/to/src -type d -printf '%P\0' | xargs -0 mkdir -p
  

• copying files with the given pattern:

    cd /path/to/src
    grep -rlZ pattern | \
        (cd /path/to/dest; \
        xargs -0 sh -c ' \
            while [ $# -ne 0 ]; do \
                grep pattern "/path/to/src/$1" >"$1"; \
                shift; \
            done' sh)
  

• restore permissions:

    rsync -a --existing /path/to/src /path/to/dest/
  

With GNU find(1), xargs(1), and grep(1):

• duplicate the directory structure:

    cd /path/to/dest
    find /path/to/src -type d ! -path /path/to/src -printf '%P\0' | xargs -0 mkdir -p
  

• restore permissions:

    rsync -a --existing /path/to/src /path/to/dest/
  

• copying files with the given pattern:

    cd /path/to/src
    grep -rlZ pattern | \
        (cd /path/to/dest; \
        xargs -0 sh -c ' \
            while [ $# -ne 0 ]; do \
                grep pattern "/path/to/src/$1" >"$1"; \
                shift; \
            done' sh)