Republican lawmakers will not be seen defending migrants.  They fear the alt-right Trump voter the same way they feared the Tea Party when they refused to cooperate with President Obama.  It is the potential primary challenger they fear more than the voter (and the libertarian donors that back them).

The Democrats are a different animal.  The last time they brought in a Churchman to do Catholic Social Doctrine, FDR was President.  I don't expect them to do so when out of power.  The Democrats are no longer the Catholic party - especially given the Bishops' focus on doing the impossible on abortion - overturning Roe and sending it back to the states (that is more a jurisdictional position than a pro-life one).  The Democrats get out the vote on this issue, so don't expect them to tell the kind of truth that makes it go away.