Here is a modern-day example. The same pawn structure as in the first example, but this time the white king is on d2. Perhaps this makes it easier for White to advance his kingside pawns, though at the same time his own king will be more vulnerable when Black succeeds in breaking through on the queenside.

Following in the footsteps of Lilienthal! The king moves towards a safer place, aiming to connect the rooks before the action starts on the queenside.

Mission completed. Black is ready to bring his king's rook into the game, while his king is safe on c7.

Here it is very probable that the battle will be fought on the kingside. Therefore Kamsky takes appropriate measures.

Thus Black has completed his artificial castling. In the meantime White is trying to create play on the queenside.

Just in time.

This allows Black to keep the queenside closed whichever pawn, a or b, White advances. This very much resembles the outcome of Petrosian's 1 8 . . . a6 ! in his 7th match game against Spassky in 1966. Now Black still has a long way to go on the kingside, but at least his king, unlike the white one, is out of the way, and he later managed to outwit his opponent.

After a bad opening, there is hope for the middle game. After a bad middle game, there is hope for the endgame. But once you are in the endgame, the moment of truth has arrived."
– Edmar Mednis.

Heading for an endgame can also be a valid reason for taking your king for a stroll. In the diagram position, the queens are very likely to disappear from the board, and because of that, Black would prefer to keep his king in the center, for instance on e7. Apparently, Anand was not satisfied with that and sent his king towards c7, where it also defends the pawn on b7.

23.Rfb1, vacating f1 for the knight, was an alternative.

It's not much of a pawn that White wins, considering his doubled c-pawns. Although he disposes of his bad bishop, it is Black who obtains the active pieces.