We propose a novel framework for exploring generalization errors of transfer learning through the lens of differential calculus on the space of probability measures. In particular, we consider two main transfer learning scenarios, $α$-ERM and fine-tuning with the KL-regularized empirical risk minimization and establish generic conditions under which the generalization error and the population risk convergence rates for these scenarios are studied. Based on our theoretical results, we show the benefits of transfer learning with a one-hidden-layer neural network in the mean-field regime under some suitable integrability and regularity assumptions on the loss and activation functions.