Try:

find . \( -name '*.mkv' -o -name '*avi' -o -name '*mp4' -o -name '*flv' -o \
          -name '*ogg' -o -name '*mov' \) -exec sh -c '
  for file do
    target="${file%.*}.mkv"
    echo ffmpeg -i "$file" "$target"
  done' sh {} +

Replace echo with whatever command you want to use.

With POSIX find:

find . \( -name '*.mkv' -o -name '*avi' -o -name '*mp4' -o -name '*flv' -o \
          -name '*ogg' -o -name '*mov' \) -exec sh -c '
  for file do
    target="${file%.*}.mkv"
    echo ffmpeg -i "$file" "$target"
  done' sh {} +

Replace echo with whatever command you want to use.

If you have GNU find or BSD find, you can use -regex:

find . -regex '.*\.\(mkv\|avi\|mp4\|flv\|ogg\|mov\)'