I just tried sshexec on my home Linux "server" and it worked now. It also automatically took my ssh key from Pageant.
Hm, I just tried to simply request a web page through the tunnel twice in a row, but apparently the browser keeps the connection open even if I close that tab. I had to wait for a timeout. for a request to fail. But closing the whole browser window rather than just the tab worked. The connection will be closed and when retrying in a new window the request will fail.