Understand what the trail actually is
Ants in your kitchen are not an invasion, they're a supply line. A scout found food, walked home, and laid down a pheromone trail. Every ant you see is following that chemical breadcrumb, and the colony itself is outside where it belongs.
This is good news. You don't need to fight thousands of ants. You need to do three things: remove the food, erase the trail, and close the door they came in through. Do all three and the traffic stops within a day or two.
Follow it both ways
Before you clean anything, spend five minutes watching. Follow the trail to what they're eating, then follow it the other way to where they're entering: a gap at the skirting, a crack by the window frame, a gap around a pipe. Mark the entry point with a bit of masking tape.
Don't clean up yet
The trail is your map. Wipe it too early and you lose the entry point, and the ants will find a new route you haven't spotted.
Remove the food and erase the trail
Deal with whatever they found: crumbs behind the toaster, a sticky honey jar, the cat's bowl. Store sweet things in sealed containers for the next fortnight. An ant colony gives up on a food source that stops paying.
Now erase the pheromone line. Mix equal parts white vinegar and water in a spray bottle and wipe the entire visible trail, entry point to food source. Washing-up liquid in warm water works too. The smell fades within an hour but the trail chemistry is gone, and followers lose the route.
Escort the stragglers
Ants still wandering after the wipe-down are lost scouts. A sheet of paper and a cup gets them outside a few at a time, or brush them gently onto a dustpan. They'll rejoin the colony.
Strong scents hold the line
Ants dislike crossing strong smells. A line of citrus peel, a wipe of peppermint oil, or ground cinnamon at the entry point works as a temporary barrier while you organise the permanent fix.
Seal the entry and keep it sealed
The permanent fix is physical. Fill the entry gap with decorator's caulk (a £3 tube and ten minutes) or, for a quick test, a smear of petroleum jelly. Check the obvious neighbours too: gaps around pipes, cracked mortar, the threshold strip by the back door.
For the next week, wipe the old trail area daily with the vinegar mix and keep surfaces crumb-free. If a new trail appears, you've found another entry point: same routine, follow, wipe, seal. Most houses only have one or two.