Some common design challenges that a potential UX Designer could face during a job interview are:
The design critique.
Here you will be asked to evaluate an app, website, service in varying level of detail. This is an exercise to showcase your understanding of design, target users, how the execution is connected to the actual problem. In general, it shows the way you think, evaluate and inquire.
Here you will be asked to complete a small task like generate a user flow, a customer journey or create a wireframe based on some given inputs. It is meant to showcase your viualization skills, quick thinking, ideation, and methodology.
Another approach is to be given a design challenge prior to the interview. This is generally a more in-depth exercise that involves all the design muscles you would use on a real project but in a more limited way. It could be to evaluate, research and redesign a real or hypothetical product.