What tasks you train with one set of features and predict with another?

Question

The most common scenario in supervised learning is to have data points with a set of features and train a model to make classification predictions afterward.

Usually, for predictions to make sense with new data points, these new data points need to have the same features and come from the same distributions.

My questions is: what are scenarios in which the data points you want to predict do not necessarily have the same set of features from the data points you trained your model with?