When I first learned about the etale fundamental group, there was a mythical theorem going around that in the algebraic case all we need to look at is the finite covers, because the infinite degree algebraic covers are inverse limits of the finite ones (obviously unlike the topological case). But I've never seen a convincing source for this theorem.
It seems reasonable that the statement would be: "every flat unramified map of a connected scheme onto a quasi-projective curve is an inverse limit of finite etale covers". Is this true? Do you have a reference for this?
