This answer provide interesting commands based on git am and presented using examples, step by step.
Objective
- You want to move some or all files from one repository to another.
- You want to keep their history.
- But you do not care about keeping tags and branches.
- You accept limited history for renamed files (and files in renamed directories).
Procedure
- Extract history in email format using
 git log --pretty=email -p --reverse --full-index --binary
- Reorganize file tree and update filename change in history [optional]
- Apply new history using git am
1. Extract history in email format
Example: Extract history of file3, file4 and file5
my_repo
├── dirA
│   ├── file1
│   └── file2
├── dirB            ^
│   ├── subdir      | To be moved
│   │   ├── file3   | with history
│   │   └── file4   | 
│   └── file5       v
└── dirC
    ├── file6
    └── file7
Clean the temporary directory destination
export historydir=/tmp/mail/dir  # Absolute path
rm -rf "$historydir"             # Caution when cleaning
Clean your the repo source
git commit ...           # Commit your working files
rm .gitignore            # Disable gitignore
git clean -n             # Simulate removal
git clean -f             # Remove untracked file
git checkout .gitignore  # Restore gitignore
Extract history of each file in email format
cd my_repo/dirB
find -name .git -prune -o -type d -o -exec bash -c 'mkdir -p "$historydir/${0%/*}" && git log --pretty=email -p --stat --reverse --full-index --binary -- "$0" > "$historydir/$0"' {} ';'
Unfortunately option --follow or --find-copies-harder cannot be combined with --reverse. This is why history is cut when file is renamed (or when a parent directory is renamed).
After: Temporary history in email format
/tmp/mail/dir
    ├── subdir
    │   ├── file3
    │   └── file4
    └── file5
2. Reorganize file tree and update filename change in history [optional]
Suppose you want to move these three files in this other repo (can be the same repo).
my_other_repo
├── dirF
│   ├── file55
│   └── file56
├── dirB              # New tree
│   ├── dirB1         # was subdir
│   │   ├── file33    # was file3
│   │   └── file44    # was file4
│   └── dirB2         # new dir
│        └── file5    # = file5
└── dirH
    └── file77
Therefore reorganize your files:
cd /tmp/mail/dir
mkdir     dirB
mv subdir dirB/dirB1
mv dirB/dirB1/file3 dirB/dirB1/file33
mv dirB/dirB1/file4 dirB/dirB1/file44
mkdir    dirB/dirB2
mv file5 dirB/dirB2
Your temporary history is now:
/tmp/mail/dir
    └── dirB
        ├── dirB1
        │   ├── file33
        │   └── file44
        └── dirB2
             └── file5
Change also filenames within the history:
cd "$historydir"
find * -type f -exec bash -c 'sed "/^diff --git a\|^--- a\|^+++ b/s:\( [ab]\)/[^ ]*:\1/$0:g" -i "$0"' {} ';'
Note: This rewrites the history to reflect the change of path and filename.
      (i.e. the change of the new location/name within the new repo)
3. Apply new history
Your other repo is:
my_other_repo
├── dirF
│   ├── file55
│   └── file56
└── dirH
    └── file77
Apply commits from temporary history files:
cd my_other_repo
find "$historydir" -type f -exec cat {} + | git am 
Your other repo is now:
my_other_repo
├── dirF
│   ├── file55
│   └── file56
├── dirB            ^
│   ├── dirB1       | New files
│   │   ├── file33  | with
│   │   └── file44  | history
│   └── dirB2       | kept
│        └── file5  v
└── dirH
    └── file77
Use git status to see amount of commits ready to be pushed :-)
Note: As the history has been rewritten to reflect the path and filename change:
      (i.e. compared to the location/name within the previous repo)
- No need to git mvto change the location/filename.
- No need to git log --followto access full history.
Extra trick: Detect renamed/moved files within your repo
To list the files having been renamed:
find -name .git -prune -o -exec git log --pretty=tformat:'' --numstat --follow {} ';' | grep '=>'
More customizations: You can complete the command git log using options --find-copies-harder or --reverse. You can also remove the first two columns using cut -f3- and grepping complete pattern '{.* => .*}'.
find -name .git -prune -o -exec git log --pretty=tformat:'' --numstat --follow --find-copies-harder --reverse {} ';' | cut -f3- | grep '{.* => .*}'