LH published new rules concerning Involuntary Schedule Changes recently:

If a flight has a schedule change, the passenger has the right to rebook the flight free of charge to another date in the same booking class on a LH operated flight. If the booking class is not available, the agent must rebook the pax to a higher booking class in the same compartment. So, you can book K and get an upgrade to Y if there are no alternatives
If this is not possible, one can rebook the flight free of charge to another date +/- 3 days from the originally booked date on a LH operated flight in the same booking class.
If this is also not suitable, one can rebook the flight free of charge to another flight on the same day on a LH codeshare, if the same booking class is available.
Even if this is also not possible, one can rebook the flight free of charge to another LH group flight (SN/LX/OS) in the same booking class.

These changes can be made by the agent without any waivers from LH. But, if the rebooking involves other booking classes, one has to ask the LH supervisors.
But, all these changes must be made within 14 days after the passenger has been informed about the time change!