How to execute a command through SSH and keep the connection open?

What you are describing is known as SSH multiplexing.

I use that setup in a devops setting for caching my connections to any VMs.

In that way I reuse the same connection for up to 30 minutes/cache the connection, without renegotiating the entire SSH connection (and authenticating the user) in each new command.

It gives me an huge boost in speed, when sending (multiple) commands in a row to a VM/server.

The setup is done on the client side, and for a cache of 30 minutes, the setup can be done in /etc/ssh/ssh_config as:

ControlPath ~/.ssh/cm-%r@%h:%p
  ControlMaster auto
  ControlPersist 30m

The MaxSessions parameter, also in ssh_config also defines how many multiplexed connections simultaneous connections are allowed; the default value is 10. If you need more simultaneous cached connections, you might want to change it.

For instance, for a maximum of 20 cached connections:

MaxSessions 20

For more information, see OpenSSH/Cookbook/Multiplexing

An advantage of SSH multiplexing is that the overhead of creating new TCP connections is eliminated.
...
The second and later connections will reuse the established TCP connection >over and over and not need to create a new TCP connection for each new SSH connection.

Also see Using SSH Multiplexing

SSH multiplexing is the ability to carry multiple SSH sessions over a single TCP connection

Without multiplexing, every time that command is executed your SSH client must establish a new TCP connection and a new SSH session with the remote host. With multiplexing, you can configure SSH to establish a single TCP connection that is kept alive for a specific period of time, and SSH sessions are established over that connection. This can result in speed increases that can add up when repeatedly running commands against remote SSH hosts.

Lastly, as the multiplexing keeps the TCP connection open between the client and the server, you will have the guarantee that you are talking with the same machine in the load balancer, as long as the cache is open/active.

There are multiple ways to do it, one you can try is

 ssh -fT -L 40000:localhost:40000 [email protected] 'sleep 60'
 FOO=$(ssh root@localhost -p 40000 cat /foo)

rest you can continue ...

Here -f is to put it into background and -T is to disable tty allocation. Using sleep command to keep session active for 60s and to allow exit itself, you can modify it or make it up by other ways.

A simple one, for example, keep the session alive until it sees a file /tmp/a.txt , as once you are done with your work, create a file , while loop will see it and run the cleanup rm /tmp/a.txt and exits the session.

ssh -fT -L 40000:localhost:40000 [email protected] 'while [[ ! -f /tmp/a.txt ]] ; do sleep 2 ; done ; rm /tmp/a.txt'
FOO=$(ssh root@localhost -p 40000 cat /foo)
do you work..
ssh root@localhost -p 40000 touch /tmp/a.txt

You can use any other conditions which suits you better to keep it alive.