We investigate trends and drivers of cross-border flows of the two major native cryptoassets (Bitcoin and Ether) and the two major asset-backed stablecoins (Tether and USD Coin) between 184 countries from 2017 to 2024. These flows are substantial, peaking at around USD 2.6 trillion in 2021, with stablecoins accounting for close to half the volume. The unique bilateral data allow us to estimate the drivers of these flows in a gravity framework, and how they differ across different types of crypto assets. Our findings highlight speculative motives and global funding conditions as key drivers of native crypto asset flows. Transactional motives play a significant role in cross-border flows for stablecoins and low-value Bitcoin transactions, where we further find a strong association with higher costs of traditional remittances. Geographic barriers play a diminished role compared to traditional financial flows, and capital flow management measures appear ineffective.