Like some here, I ended up writing a wrapper around the JSch library.
It's called way-secshell and it is hosted on GitHub:
https://github.com/objectos/way-secshell
// scp myfile.txt localhost:/tmp
File file = new File("myfile.txt");
Scp res = WaySSH.scp()
.file(file)
.toHost("localhost")
.at("/tmp")
.send();