If you're being really clever about it, you don't even need to use a backup file at all, on the Linux side, you can simply do:
mysqldump -h windowsbox -u username -p databasename | mysql -u username -p databasename
Which uses a pipe to connect the dump to the mysql client.
In order for this to succeed you must have at least read and lock access to the database you're backing up, and create access to the new one, which must already exist and not already contain tables of the same names.
Mark