The "cornerstone" is that the federal government should be limited to powers enumerated in The Constitution. That does include enforcing our national border. It does not include things like education and abortion -- and those things are thus state-level issues.
Sanctuary jurisdictions cannot legally stop the federal government from removing illegal aliens (side note: these people are not "immigrants"). That doesn't mean places like LA won't be successful obstructionists, at least to some extent. But if the feds want to get nasty about it, well, sanctuary jurisdictions are in for a rough ride.
The sight of federal police exercising power in jurisdictions adamantly opposed to it could easily morph into civil war, IMO, so both sides need to tread carefully.