Italy started searching for migrants at sea after the Lampedusa disaster, in which at least 300 migrants drowned when a migrant ship sank travelling to Europe. These Italian search and rescue operations have been called 'Mare Nostrum', meaning 'Our Sea' in Latin.
The Italian government has said it is reducing the number of search and rescue operations it undertakes, and the UK has said it will not take Italy's place.
Bishop Patrick Lynch, chair of the Office for Migration Policy at the Catholic Bishops' Conference Department of International Affairs, said: "Last summer at Lampedusa, Pope Francis challenged us to take responsibility for our brothers and sisters.
"He warned us against losing our bearings in a culture of comfort and becoming insensitive to the cries of other people.
"Since the tragedies of last October, the Italian navy has led efforts to save many thousands of lives but now this is a time for all European governments to recognise that others' suffering is our business.
"While efforts are being made to find lasting solutions to these challenges, we have a duty to heed with compassion the cries of our wounded brothers and sisters, and not to pass by on the other side.
"Britain is still Europe's leading naval power, and to refuse to join in Mediterranean search and rescue operations would be a misguided abdication of responsibility to those thousands of men, women, and children who have been driven from their homes by persecution and war and forced to risk death at sea."
The Home Office has responded, stating: "Ministers across Europe have expressed concerns that search and rescue operations in the Mediterranean have acted as a pull factor for illegal migration, encouraging people to make dangerous crossings in the expectation of rescue.
"This has lead to more deaths as traffickers have exploited the situation using boats that are unfit to make the crossing.
"At the last JHA [Justice and Home Affairs] council, ministers agreed that the operations should be brought to a well-publicised end, with enhanced operations by the EU's External Border Agency, Frontex, introduced closer to the EU's maritime border.
"The UK has already offered initial support for this enhanced Frontex operation, and we are considering a further contribution.
"The UK and its European partners have also agreed to enhance operational co-operation within the EU and to work with countries of origin to tackle the causes of illegal immigration and the organised gangs that facilitate it and to enhance support for protection in North and East Africa for those who need it."
This year has seen a significant increase in migrant boats coming to Europe and it is reported that 3,000 people have drowned.
Frontex will be running a project called Operation Triton in place of Italy's Mare Nostrum search and rescue mission.
Operation Triton focuses on border control and does not have a search and rescue function like Mare Nostrum's, although the Italian navy says it will still carry out some operations despite only having a third of the budget.