Many have been intercepted, with 12 men from Syria and Iran brought to shore at Dover on Friday and 40 migrants detected on Christmas Day.
The main United Kingdom opposition leader Jeremy Corbyn said there was a "duty to reach out the hand of humanity" to people facing danger and seeking safety.
Against a backdrop of speculation over senior Tories positioning themselves to succeed Theresa May, Defence Secretary Gavin Williamson said the armed forces were ready to offer help if needed - but there had been no requests from the Home Office.
Conservative ministers have been condemned for "aping Donald Trump" by using the issue of migrants attempting to cross the English Channel for political gain. "The scale of the problem is unprecedented", Charlie Elphicke, the MP for Dover on the southeast coast of England, told the Daily Mail.
Six Iranian men were found on a Kent beach early on Sunday, in the latest successful crossing of the Channel from France by migrants in inflatable boats.
Officers attended to assist Border Force officials who are dealing with the case.
Sajid Javid is to chair talks in London later after agreeing a joint action plan with the French interior minister.
The Home Office did not give information on how this would be done.
Mr Elphicke said: "I welcome the Home Secretary taking personal responsibility for the situation".
"Our focus needs to be on much better cooperation with the French authorities, and European nations as a whole to create safe and legal routes for those seeking asylum", Abbott added.
She added: "The co-operation between us and the French has been very effective and is ongoing. I am calling for our Border Force cutters now cruising the Mediterranean to return to United Kingdom waters and form a new Dover Patrol to take back control of our borders immediately".
Twenty three migrants were involved in three separate rescues on the Kent coast overnight and this morning.
Davey said the Lib Dems want more investment in a Stronger Border force to tackle criminal gangs trafficking people, drugs and weapons and undo the "damage done by Theresa May's cuts to Britain security".
He warned that more boats might act as a magnet for migrants hoping to be picked up and taken to the UK.
The UK's National Crime Agency says French authorities have disrupted numerous attempts to make the perilous journey.
Mr Javid said yesterday: 'The situation in the Channel is of grave concern, with people gambling their lives in reckless attempts to reach the United Kingdom in unsafe boats and treacherous conditions'.