For those is the constituencies of Southern England, I believe the answer is yes. The people in the Thames valley and further south were disillusioned by what (to them) seemed the failure of Tory policy on Black Wednesday, and beset by negative equity. They blamed the Tories I doubt John Smith would have gone as far to pander to the Tories as Blair did, but we shall never know. I do know that Roy Hattersley, hardly a left wing fire brand, found the Blair project distasteful.
The main message in 1997 was about hope. Huge investment in the NHS. Minimum wage. Education funding increases. And so on. And they delivered. 2001 is when Blair's god complex started in earnest.
I think there has to be a balance of justice test somewhere. If you're genuinely likely to suffer torture or death by being sent home then that's different but there are plenty of genuine cases where families are split up because they've innocently tripped one of the bureaucratic nonsensities and one parent has been deported.
Even then, my very libertarian views are tested by people coming here and committing crimes then demanding that we honour their rights.