You can use System.IO classes like File to copy files
File.Copy Method (System.IO)[
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create shares on the target\destination servers and copy the files between them, so from "\\servera\myshare\myfile.txt" to "\\serverb\myshare\myfile.txt" or whatever. Your biggest issue is going to be security, the account your .net code runs under will need access to all of these shares and by default the account used won't. You'll need to create an account with the desired access and configure IIS to use that account rather than the default one.
Configure the Anonymous Authentication Identity (IIS 7)[
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Or you can leave it running as the app pool user and change the user the app pool uses (try saying that 10 times fast).