I was tasked with migrating several aging and nested distribution groups into one large group. Even though Windows Powershell is horrible, it was still a relatively easy process.
 
Dump existing group members to CSV file.

Get-DistributionGroupMember -identity "Group1" | Export-Csv C:members.csv

Then import the members and add them to the new group. In my case, I had already created the group.

Import-Csv C:members.csv | foreach {Add-DistributionGroupMember -Identity "NewGroup" -Member $_.displayname}