Hi!
Had the same problem. Here I have read, that cf7 adds some scripts, when wp_footer() is called in the theme.
I looked and saw, that these scripts names were jquery.form.js and scripts.js . I tried to find, where exactly submit behaviour was bound, but unsuccessfully. So after every ajax refresh I just added these lines (.container is the block with updated content):
$('.container').append('<script src="...jquery.form.js"></script>');
$('.container').append('<script src="...scripts.js"></script>');
