If the source phone is rooted you have multiple possibilities.
If the target phone is also rooted one easy solution may be a back as it is described in Is there a way to do Nandroid backup directly to pc and then restore it directly from pc also?
First connect via
adb shell to the phone and get root permissions via
Now create backup all app files via tar:
tar -cvf /sdcard/app-data.tar /data/data
Then exit adb and copy the created file to your computer:
adb pull /sdcard/app-data.tar
Afterwards you should have a file named app-data.tar on your computer. Check it with an archive tool and you will find a lot of directories.
You may want to delete certain files or directories from the tar file.
Afterwards you can convert it to an Android Backup so that you can write back the data onto the new un-rooted phone.
Making a tar file to an Android Backup can be done via Android Backup Extractor. Note that some apps will refuse to accept the restored files (because backup is disabled).
The created Android backup can then be restored via
adb backup command.