This is a recent Dutch expression, but I don’t manage to find out where or by who it was used first.

Where does the phrase "It runs like a tit" come from?
"It runs like a tit" means something is progressing.

Tit does not refer to a woman's breast in this saying, as is often thought, but to a chicken. In different dialects (including in Breda, Utrecht and Goeree - Overflakkee) tit means "chicken, hen’. Walk like a tit literally means ' run fast '. It can still be used in that sense, although the more figurative meaning is more common now. The word tit, according to the Dictionary of the Dutch language is in those dialects an onomatopoeia, an imitative word. Chickens were lured by the sound tit - tit - tit. That sound was then used to identify the animals themselves.

Because of the association with 'woman's breast' is run like a tit sensed as a rather informal phrase. Neutral alternatives are run like a train.

The possible origin should still be sought elsewhere, namely tit as the Flemish dialect word Tittie (titje) or titsie (titske). This means "a piece, trifle, little '. And would have been used for a (little, strong) drink.

"As a tit 'can be combined with running and selling. The expression "Walk like a tit 'is listed in the Dictionary of the Dutch language, in an article written in 1941 (tit-tit here is the lure-sound). 'Tit' is a Dutch dialect word for 'chicken'. "Run like a tit" means "Run like hell." Later it got the figurative meanings.

The Dutch TV presenter Jan Lenferink said in 1988 Humo that his TV show ran like a tit. Marc De Coster took the phrase in his Dictionary of Neologisms in 1999.
But who said first "Swinging like a tit?

The correctness of the statement seems to me to be highly dependent on the cup size.