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Far from done with it.

Do you happen to know if it's possible to transfer tokens on the Ethereum chain? Tokens like babydoge and other 💩 coins?

Yeah, the main Swap uses Ethereum and other EVM chains like BSC, Polygon, Arbitrum, etc, and uses all decentralized liquidity from Uniswap, Sushiswap, etc.

And yes on the cross chain swap as well. If you are trying to buy cross chain, use the Simpleswap integration as those coins should be available.