Authorities were called to five separate instances involving people presenting as Iraqi, Iranian and Afghan, the Home Office said.
The Christmas Day crossings follow a number of migrant rescues in recent months involving people trying to travel across the Channel to Britain.
A Home Office spokeswoman said: "Border Force has intercepted a number of boats containing migrants attempting to cross the Channel overnight and this morning, and has deployed resources to deal with these incidents.
"The evidence shows there is organised criminal gang activity behind illegal migration attempts by small boats across the Channel.
"We are working closely with the French and law enforcement partners to target these gangs, who exploit vulnerable people and put lives at risk."
A girl was among eight people who arrived in Folkestone, Kent, at around 2.40am on Christmas Day.
A child was also one of 13 people on board a dinghy assisted by Border Force off the coast of Deal, Kent, shortly before 7am.
All of those involved underwent a medical assessment and the adults have been transferred to immigration officials to be interviewed.
The children will come under the care of social services, officials said.
In another incident in French waters, nine migrants were rescued and are now in the UK after the engine of the vessel they were travelling in failed.
Border Force officials are being assisted by the Maritime and Coastguard Agency after being called to an incident in the Channel before 9.20am involving two people in a small boat travelling towards the UK.
All 40 people are now in the UK, the Home Office said.