Animal Rebellion says more than 100 people climbed on to trucks, milk silos and loading bays as part of its campaign for a plant-based food system.
The Muller and Arla sites in Buckinghamshire, Gloucestershire, Somerset and Worcestershire were targeted from about 05:00 BST.
Muller and Arla said milk supplies would continue.
Police are at all four sites and have made some arrests.
Muller operates sites in Droitwich in Worcestershire, Severnside in Gloucestershire and Bridgwater in Somerset.
Arla, which runs the UK’s largest milk factory in Aylesbury in Buckinghamshire, has a recently-obtained injunction in place that aims to stop people protesting at its sites.
Animal Rebellion said it was taking the action after receiving no response to a letter delivered to Downing Street in August, warning of disruptive action this month unless progress towards its demands was made.
The group said it wanted “wholesale governmental support for farmers and fishing communities to transition to a plant-based food system and programme of rewilding that will secure a future for generations to come”.
More protests are planned unless negotiations take place with the government, the group said.
“The goal here is to make a national crisis and stop the supply of dairy to supermarkets,” said Joel Scott-Halkes, from the group.
“We’re doing this because we want the government to stop pouring billions of pounds of taxpayers’ money into the failing, toxic and polluting industry that is animal farming that’s driving our planet to the point of collapse.”
Thames Valley Police said it was made aware of a protest at Arla Dairy Aylesbury at around 05:50 and had so far made five arrests.
Buckinghamshire Council said both slip roads off the nearby A41 at College Road North had been closed by police, meaning there was no access to the waste and recycling centre there.
Ch Insp Stuart Bell, from Avon and Somerset Police, said seven people accessed the site just after 06:00 and several others were stopped. One person has been arrested.
“Specialist policing resources are being brought in to assist with this operation as the protesters remaining within the plant are either at height or are locked on to something,” he said.
“While some operations at the dairy have been affected, there’s no impact on the wider public and no disruption to surrounding roads.”
West Mercia Police said officers were called to the Droitwich site after reports that 20 people had gained unauthorised access.
By 15:30, 13 people had been arrested for aggravated trespassing offences.
Gloucestershire Police said police seized equipment designed to cause serious disruption and five people had been arrested.
The force “upholds the right of individuals to protest in a peaceful manner, but will take action against anyone who commits any unlawful act against either people or property”, a spokesperson said.
In a statement, Muller said: “We are disappointed to be targeted by a small number of activists who don’t represent the 96% of adults in Britain who choose milk every week, and we will ensure that supplies are maintained.
“Dairy is affordable and packed with nutrients that benefit our bodies. During a cost of living crisis it is wrong to try to prevent it from reaching families, including vulnerable members of society.”
A spokesperson for Arla said: “We currently have protesters at our Aylesbury site and are working closely with the local police to resolve the situation.
“The safety and security of our colleagues at the site is our number one priority and production is currently running as normal.”
The Department for Environment, Food and Rural Affairs has been asked for a comment.
