Can someone educated me regarding LH rebooking rules?

Ticket was a business class award from United on intra-Europe flights on LH metal. Flights XXX-FRA-YYY. XXX-FRA was delayed and we missed FRA-YYY. This required an overnight in FRA and rebooking into next day FRA-YYY.

When I called LH after receiving a “contact service center for rebooking FRA-YYY” message, the LH service center refused to rebook the ticket and told me I had to call United Mileage Plus (LH had obviously taken operational control of the ticket by then). In FRA, a LH agent effortlessly changed the tickets and rolled his eyes when I mentioned the ‘call United’ statement.

It turned out LH had automatically rebooked us, but downgraded us from business into economy. The LH agent reseated us in businesses (6/36 seats taken with 4 of these us, so wide open). Final point: LH changed the ticket class to “K”, deepest discount economy (seats were still in business).

Can someone comment on this? What are rebooking rules once LH takes operational control, and was the conversion to K accurate?

Thanks.