We develop a theory of transfer learning in infinitely wide neural networks under gradient flow that quantifies when pretraining on a source task improves generalization on a target task. We analyze both (i) fine-tuning, when the downstream predictor is trained on top of source-induced features and (ii) a jointly rich setting, where both pretraining and downstream tasks can operate in a feature learning regime, but the downstream model is initialized with the features obtained after pre-training. In this setup, the summary statistics of randomly initialized networks after a rich pre-training are adaptive kernels which depend on both source data and labels. For (i), we analyze the performance of a readout for different pretraining data regimes. For (ii), the summary statistics after learning the target task are still adaptive kernels with features from both source and target tasks. We test our theory on linear and polynomial regression tasks as well as real datasets. Our theory allows interpretable conclusions on performance, which depend on the amount of data on both tasks, the alignment between tasks, and the feature learning strength.