Mostly, Leftists will not listen to - leave aside engage with - evidence and rational argument from Christians.
But since the development of New Left political correctness, and its spread to include the almost whole of the ruling class and most of everyone else, there is a definite sense that argument with modern progressives is counter-productive.
I think the reason for this is quite simple: inversion.
For the modern Left, to a striking extent, the evaluative framework is an inversion of the traditional; so Good is bad - and bad is Good.
This applies to aesthetics, to standards of truth, and to morality.
Typically, the Christian or conservative or libertarian puts forth a consequentialist argument on the lines of:
'If this [insert currently promoted Leftist policy] is implemented - then the following [insert bad consequences] will ensue...'
But - because of inversion - what count as 'bad consequences' for the Right, count as good consequences for the modern Left.
So if the evidence and reason suggest that policy X will result in damage to marriage, the decline of the family, an increase in violence and disorder, or encouragement of parasitic behaviour - then what the Left hears is that the policy will work in exactly the way they want it to work.
Christians arguing with the PC ruling class of modern Western societies are therefore, but unwittingly, in a position analogous to that of a good German in 1933 telling the Nazis that both evidence and reason suggest that their policies will probably harm Jews.
The imaginary good German of 1933 neglects that the Nazis were anti-semitic, and that therefore their policies were intended to harm Jews - the Christian or Right-wing or libertarian mainstream commentator neglects that the modern dominant Left is anti-the Good and therefore their policies are intended to harm the Good.
And, just as the Nazis originally wanted several Good things as well as the destruction of Jews, but as time went by the destructive agenda became ever more dominant at the cost of any Good things; so modern Leftists have moved-on from the time when they wanted several Good things alongside harming their enemies, to an ever-more-complete embrace of their destructive agenda.
The Nazis started off pro- some Good things as well as anti-semitic but by 1945 had ended-up mostly anti-semitic; the Left is now at a similar end-point when whatever Goods the Left used to favour are being swallowed-up and lost in their desire to destroy that which they hate.
Christians argue from the assumption that the modern ruling elite want to improve things in a functional sense - want to reduce poverty or unemployment, want to improve the life of Africans, want to protect the environment, want to improve health services or education, want to have a more just legal system, more efficient research, a more effective military; want a peaceful and wholesome society...
But the ruling elite want none of these things, or at least if they personally happen to want one of them, then this individual idiosyncracy is constrained by the imperatives of their over-arching ideology.
As a class what our rulers want is mostly to destroy what they hate.
When the rulers in the mass media, politics, public administration, law and education are told - evidentially and with good reasoning - that their policies will likely cause damage to things that traditional Christians (and indeed those with common sense and who take seriously their personal experience) regard as Good; what the ruling elites are actually hearing is that their policies will damage the kind of people and situations that Leftists regard as bad; they are hearing confirmation that these are the correct policies.
Thus the bulk of mainstream mass media Right Wing and Christian commentary serves the Left as a test and confirmation of the correctness of their policies, and a measure of the optimal priorities.
The more conclusive is the Right wing and religious assembling of evidence and rational argument against a proposed Leftist policy; the more aggressively that policy will be promoted because the Left then know - to a high degree of probability - that the policy will achieve its desired effect.