(CNN) -- Known for his outspoken, unapologetic support of migrants in Mexico, the Rev. Alejandro Solalinde is bringing his message to the United States.

The priest is part of a caravan of migrants and their supporters traveling from Los Angeles to Washington to push for immigration reform.

In Mexico, Solalinde has criticized the government, and even the Catholic Church, saying that both can be more compassionate to migrants. His views are shaped by the years he has spent leading a migrant shelter in Oaxaca that offers support to Central Americans who embark on the dangerous route north by clinging to trains.

A number of threats last year led to his leaving his post, located in Ixtepec, in the southern state of Oaxaca, but he has since returned.

"I don't know how to live with fear," Solalinde told CNN.

Immigration issues must be tackled both at the source and the destination of the migrants, he said.

In the United States, Solalinde says he wants to include more voices in the debate. "Not just the voice of the north, but the voice of the south, the voice of the poor, the voice of the migrants."

The 68-year-old, wearing a white button-down shirt, cream-colored vest, thin glasses and a cross necklace, admits that he believes his opinion will have little sway. With so much money and political interests at play, the voice of a Mexican priest is likely to get drowned out, he said.

But Solalinde is undeterred. The caravan's participants are lobbying for a path to citizenship, a stop to the separation of families, and greater attention to the human cost of immigration policies. The priest says he is not advocating a certain policy, but wants to raise consciousness about the issue.

An immigration bill currently pending in the Senate commits additional resources to southern border security and establishes a new system of metrics to measure border control effectiveness, but critics insist the plan is full of holes and will ultimately do little to help stem to the tide of illegal immigration.

While claiming to strengthen the border, the legislation also would create a 13-year path to citizenship for most undocumented immigrants while bringing dramatic change to labor policy on America's farms.

U.S. tactics like a border wall and restrictive immigration policies do nothing other than drive up the price that smugglers charge to sneak people across the border, he said.

At the migrant shelter in Oaxaca, Solalinde says he sees constant suffering, but also constant hope in the people who pass by.

During his brief exile from the shelter, Solalinde spent a month in Europe followed by some weeks of reflection in Jalisco state before returning to Oaxaca.

After doing much reading and writing during his time away, the priest said he returned with a deeper commitment for acting now, for helping migrants today, he said.

"Each day god gives me the grace to be happy, he provides strength, but at the end of the day my heart hurts, it tires emotionally," he said.

CNN's Alan Silverleib contributed to this report.