Though this seems like an fairly easy requirement it turned out not to be something that could be done with Backup-SPSite and Restore-SPSite code, so I developed the code below, which works in a test environment I created.
Add-PSSnapin Microsoft.SharePoint. Powershell
$oldm = Get-SPuser -Web "http://win-l31vicrc3b5/" -Group "Team Site Members"
$oldo = Get-SPuser -Web "http://win-l31vicrc3b5/" -Group "Team Site Owners"
$oldv = Get-SPuser -Web "http://win-l31vicrc3b5/" -Group "Team Site Visitors"
Restore-SPSite "http://win-l31vicrc3b5/" -Path C:\backup\back.back -Force
$newm = Get-SPuser -Web "http://win-l31vicrc3b5/" -Group "Team Site Members"
$newo = Get-SPuser -Web "http://win-l31vicrc3b5/" -Group "Team Site Owners"
$newv = Get-SPuser -Web "http://win-l31vicrc3b5/" -Group "Team Site Visitors"
foreach ($spuser in $newm){Remove-SPUser -Web "http://win-l31vicrc3b5/" -Identity $spuser -Group "Team Site Members" -confirm:$false}
foreach ($spuser in $newo){Remove-SPUser -Web "http://win-l31vicrc3b5/" -Identity $spuser -Group "Team Site Owners" -confirm:$false}
foreach ($spuser in $newv){Remove-SPUser -Web "http://win-l31vicrc3b5/" -Identity $spuser -Group "Team Site Visitors" -confirm:$false}
foreach ($spuser in $oldm){Set-SPUser -Web "http://win-l31vicrc3b5/" -Identity $spuser -Group "Team Site Members" -confirm:$false}
foreach ($spuser in $oldo){Set-SPUser -Web "http://win-l31vicrc3b5/" -Identity $spuser -Group "Team Site Owners" -confirm:$false}
foreach ($spuser in $oldv){Set-SPUser -Web "http://win-l31vicrc3b5/" -Identity $spuser -Group "Team Site Visitors" -confirm:$false}
write-host "Done"
$oldm = Get-SPuser -Web "http://win-l31vicrc3b5/" -Group "Team Site Members"
$oldo = Get-SPuser -Web "http://win-l31vicrc3b5/" -Group "Team Site Owners"
$oldv = Get-SPuser -Web "http://win-l31vicrc3b5/" -Group "Team Site Visitors"
Restore-SPSite "http://win-l31vicrc3b5/" -Path C:\backup\back.back -Force
$newm = Get-SPuser -Web "http://win-l31vicrc3b5/" -Group "Team Site Members"
$newo = Get-SPuser -Web "http://win-l31vicrc3b5/" -Group "Team Site Owners"
$newv = Get-SPuser -Web "http://win-l31vicrc3b5/" -Group "Team Site Visitors"
foreach ($spuser in $newm){Remove-SPUser -Web "http://win-l31vicrc3b5/" -Identity $spuser -Group "Team Site Members" -confirm:$false}
foreach ($spuser in $newo){Remove-SPUser -Web "http://win-l31vicrc3b5/" -Identity $spuser -Group "Team Site Owners" -confirm:$false}
foreach ($spuser in $newv){Remove-SPUser -Web "http://win-l31vicrc3b5/" -Identity $spuser -Group "Team Site Visitors" -confirm:$false}
foreach ($spuser in $oldm){Set-SPUser -Web "http://win-l31vicrc3b5/" -Identity $spuser -Group "Team Site Members" -confirm:$false}
foreach ($spuser in $oldo){Set-SPUser -Web "http://win-l31vicrc3b5/" -Identity $spuser -Group "Team Site Owners" -confirm:$false}
foreach ($spuser in $oldv){Set-SPUser -Web "http://win-l31vicrc3b5/" -Identity $spuser -Group "Team Site Visitors" -confirm:$false}
write-host "Done"