To former President Donald J. Trump and his running mate, the root cause of many of the issues in the United States is simple.

Be it gun violence, high housing costs, long wait times at emergency rooms or an impending depletion of disaster relief funding, Mr. Trump and Senator JD Vance have offered the same diagnosis: All are because of unauthorized immigration.

“The mass migration invasion has crushed wages, crashed school systems — your systems are a disaster — wrecked the standard of living and brought crime, drugs, misery and death,” Mr. Trump said at a rally in California over the weekend.

By casting migrants as agents of ruin and destruction, the Republican nominees are peddling a raft of specious claims that underline the extent to which nativist sentiment has become central to their campaign. But seldom is unauthorized immigration the actual cause of the problems they say plague the country.