MySQL 5.7.4 now supports Transportable Tablespaces for InnoDB Partitions.

When InnoDB transportable tablespaces was introduced in 5.6 it did not support partitions since it could be accomplished in combination with EXCHANGE PARTITION. To make it easier to use transportable tablespaces for partitioned tables we added support for DISCARD/IMPORT TABLESPACE for partitioned tables as well as single partitions in 5.7.4.To get a consistent full copy of the tablespace files one use FLUSH TABLES <list of tables> FOR EXPORT: 

# Flush all dirty pages to the tablespaces and write a .cfg file per tablespace
mysql> FLUSH TABLE partitioned_table FOR EXPORT;
# Copy the .ibd and .cfg files and optionally the .frm files
# If only interested in a subset of the partitions,
# use 'tablename#P#{p1[,p2[...]]}' instead of tablename below.
$ cp /path/to/mysql-datadir/db-name/partitioned_table#*.{ibd,cfg} /path/to/backup/db-name/
# Release the tablespace locks
mysql> UNLOCK TABLES;

To restore the all the partitions (optionally on another server):

# If the table does not exist or has a different definition:
mysql> CREATE TABLE partitioned_table ;
# Discard the tablespaces
mysql> ALTER TABLE partitioned_table DISCARD TABLESPACE;
# Copy the tablespace files
$ cp /path/to/backup/db-name/partitioned_table#*.{ibd,cfg} /path/to/mysql-datadir/db-name/
# Import the new tablespaces
mysql> ALTER TABLE partitioned_table IMPORT TABLESPACE;

To restore only a subset of the partitions do:

# If not the table already exists, create it
mysql> CREATE TABLE partitioned_table ;
# Discard the tablespaces for the partitions to be restored
mysql> ALTER TABLE partitioned_table DISCARD PARTITION p1,p4 TABLESPACE;
# Copy the tablespace files
$ cp /path/to/backup/db-name/partitioned_table#P#p{1,4}.{ibd,cfg} /path/to/mysql-datadir/db-name/
# Import the tablespaces
mysql> ALTER TABLE partitioned_table IMPORT PARTITION p1,p4 TABLESPACE;

So in 5.7 it will be as easy to handle partitions and partitioned tables as for non-partitioned InnoDB tables.

But how to do this in 5.6?

FLUSH TABLES <list of tables> FOR EXPORT also works for partitioned tables in 5.6, so the above commands for FLUSH TABLES FOR EXPORT in 5.7 can be used in 5.6 as well for copy/backup purpose.

But to restore a partition in 5.6 one need to use EXCHANGE PARTITION for each partition:

# Create a non-partitioned table like the partitioned table
mysql> CREATE TABLE tmp_table LIKE partitioned_table;
mysql> ALTER TABLE tmp_table REMOVE PARTITIONING;
# Discard the tablespace
mysql> ALTER TABLE tmp_table DISCARD TABLESPACE;
# Copy the tablespace files
$ cp /path/to/backup/db-name/partitioned_table#P#p1.cfg /path/to/mysql-datadir/db-name/tmp_table.cfg
$ cp /path/to/backup/db-name/partitioned_table#P#p1.ibd /path/to/mysql-datadir/db-name/tmp_table.ibd
# Import the tablespace into the tmp_table
mysql> ALTER TABLE tmp_table IMPORT TABLESPACE;
# restore the partition from the tmp_table
mysql> ALTER TABLE partitioned_table EXCHANGE PARTITION p1 WITH TABLE tmp_table;
# Goto DISCARD step to restore another partition
# Clean up
mysql> DROP TABLE tmp_table;

Notes:

• If done concurrently as other DML/DDL’s happen I recommend LOCK TABLE or other tools guarding against other sessions either alter the table or alters the data in between the statements. Also verify that the supposed empty tmp_table did not get any new data in between the statements.
• EXCHANGE PARTITION currently scans the table to replace the partition to validate that every row will fit into the partition, I hope to address this by adding a ‘WITHOUT VALIDATION’ clause to the EXCHANGE command, to allow the DBA to directly exchange the data without scanning every record. IMPORT PARTITION TABLESPACE does not validate the data to match the partition.