Excluding Basic Economy, UA has a blanket change-fee waiver in place for direct-booked tickets originating from the US. ( https://www.united.com/ual/en/us/fly...hange-fee.html ). LH does not, meaning that LH will follow whatever's in the underlying fare rules.

Agency-booked tickets will follow the agency's policy, constrained by the airline's fare rules. UA's no-change-fee policy does not apply for agency-booked tickets. Instead, most nominally nonrefundable tickets give you two options: you can rebook to a higher-priced ticket and pay the fare difference, or you can get a refund to your original form of payment, less a fee. Note that your travel agent may charge an additional fee on top of whatever UA charges.

This will all be spelled out in the fare rules. However, note that if your ticket is comprised of multiple fares, the most restrictive rules apply to the entire journey.

TL/DR: book directly with UA for the most flexibility on itineraries originating in the USA.