Fedimint uses something called a LN gateway. This is a LN node operator that runs a specific plugin/daemon alongside their node and can offer their liquidity to the fedimint. They're essentially a specialized client of the federation that deals in ecash<->LN contracts. If a LN invoice is given for internal to the mint, it just settles out ecash and the LN isn't touched.

I can get a bit more technical if you're LN tech savvy. or join the Discord and pop into #lightning to chat (chat.fedimint.org)