Little is known about what makes crosslingual transfer hard, since factors like tokenization, morphology, and syntax all change at once between languages. To disentangle the impact of these factors, we propose a set of controlled transfer studies: we systematically transform GLUE tasks to alter different factors one at a time, then measure the resulting drops in a pretrained model's downstream performance. In contrast to prior work suggesting little effect from syntax on knowledge transfer, we find significant impacts from syntactic shifts (3-6% drop), though models quickly adapt with continued pretraining on a small dataset. However, we find that by far the most impactful factor for crosslingual transfer is the challenge of aligning the new embeddings with the existing transformer layers (18% drop), with little additional effect from switching tokenizers (<2% drop) or word morphologies (<2% drop). Moreover, continued pretraining with a small dataset is not very effective at closing this gap-suggesting that new directions are needed for solving this problem.