The meaning "to cause" used to belong to the phrase «влечь за собой»: «Преступление влечет за собой наказание». In the Russian corpus, the first use without "за собой" is in 1975. Apparently we're not used to using «влекомый» with this freshly discovered meaning yet.
«Наказание, влекомое за собой преступлением» sounds unwieldy but could - says my linguistic intuition - appear in some legal or philosophical text, where the unwieldier the language the higher the score.
Ok, extra analysis:
The first, main, meaning of «влечь» - to drag - is spatial: X drags Y, making Y to follow itself (i.e. move behind it in space).
The second, originally figurative, meaning - to cause - is temporal, based on a metaphorical reinterpretation of time as space: X cases Y, making Y to follow itself (i.e. happen after it in time).
Now, «тележка, влекомая лошадкой» uses present tense, placing both a horsy and a cart in the same moment: right now the car is being dragged by the horse.
«Наказание, влекомое преступлением» attempts to do the same - to place crime and punishment in the same, present, moment. But that's not how causality works, at least on macro level, so our brain/intuition resists.
To prove this, I would need to construct a phrase in such a way that there is a sense of temporal sequence in it, using passive participle in future or past tense. But when I try to conjugate «влекомый» to either it declines.
In fact, your second example displays yet another shade of meaning - to attract: «магнит влечёт железяки / железяки, влекомые магнитом» (unusual for modern language but cf. «Известно любопытным естеств взыскателем есть, что магнит камень силу свою, которою влеком есть к железу, окормляет, силою железа себе обложеннаго»), «запах влечёт мух / мухи, влекомые запахом» (sounds more natural) and «интерес к искусству влечёт человека / человек, влекомый интересом к искусству». This meaning is again figurative, based on reinterpretation of the described phenomena as if there was some sort of virtual rope with which magnets, smells and interest in arts drag correspondingly iron particles, flies and men. Since attraction happens with agent and patient both being present, the phenomena are again compatible with present tense participle.