Public figures receive disproportionate levels of abuse on social media, impacting their active participation in public life. Automated systems can identify abuse at scale but labelling training data is expensive and potentially harmful. So, it is desirable that systems are efficient and generalisable, handling shared and specific aspects of abuse. We explore the dynamics of cross-group text classification in order to understand how well models trained on one domain or demographic can transfer to others, with a view to building more generalisable abuse classifiers. We fine-tune language models to classify tweets targeted at public figures using our novel DoDo dataset, containing 28,000 entries with fine-grained labels, split equally across four Domain-Demographic pairs (male and female footballers and politicians). We find that (i) small amounts of diverse data are hugely beneficial to generalisation and adaptation; (ii) models transfer more easily across demographics but cross-domain models are more generalisable; (iii) some groups contribute more to generalisability than others; and (iv) dataset similarity is a signal of transferability.