Storage disaggregation underlies today's cloud and is naturally complemented by pushing down some computation to storage, thus mitigating the potential network bottleneck between the storage and compute tiers. We show how ML training benefits from storage pushdowns by focusing on transfer learning (TL), the widespread technique that democratizes ML by reusing existing knowledge on related tasks. We propose HAPI, a new TL processing system centered around two complementary techniques that address challenges introduced by disaggregation. First, applications must carefully balance execution across tiers for performance. HAPI judiciously splits the TL computation during the feature extraction phase yielding pushdowns that not only improve network time but also improve total TL training time by overlapping the execution of consecutive training iterations across tiers. Second, operators want resource efficiency from the storage-side computational resources. HAPI employs storage-side batch size adaptation allowing increased storage-side pushdown concurrency without affecting training accuracy. HAPI yields up to 2.5× training speed-up while choosing in 86.8% of cases the best performing split point or one that is at most 5% off from the best.