What, precisely, is kamma/karma? In Buddhist doctrine there is always the problem, that for the sake of understandability, certain concepts are being presented in a solid, substantial way, while actually, one has to keep in the back of one's mind, that by the central tenet of anattā/anātman these concepts are actually not like that. The same applies to kamma/karma, which can be presented as something that a being accumulates and carries with itself in this life and to the next and the ones following.

Now, actually, there is nothing substantial that transmigrates from one life to the next, as is clear from the similes that are used to explain rebirth without soul: a candle that lights another, an echo, a mirror image, the imprint in wax of seal. In these cases, though there is some link between the two sides, nothing actually goes from one to the other (the case with the echo should maybe be reconsidered according to the old Indian theories of sound and the echo). So, what is kamma/karma? How does it stick to the individual?

To take this question one step further, if we admit now that kamma/karma is near-identical with the concept of saṃskāra/saṃkhāra, and therefore with volitions and cetanā, as in this quote from AN 6.63:


  cetanāhaṃ bhikkave kammaṃ vadāmi
  
  Intention, I tell you, is kamma


and that by karmic unwholesome action the consciousness "leans towards" unwholesomeness, then how are these leanings supposed to look, how are they thought to be transported from one conscious moment to the next?
Karma is not something one accumulates; this question gets asked so often because, as you say, there is a misconception of it being a 'a solid, substantial' entity.

Karma is volition (intention is a bad translation, IMO), or in abhidhamma, the seven javana citta present in an ordinary mind process (i.e. every moment of experience).

All this means is that every experience you have has the potential to affect future experience in some way; the fact that karma produces a result in kind is indicative of how natural the process is - karma is simply the law of causality that moulds our lives according to our actions. That this should continue after death is not a problem for Buddhism since, as I've noted several places here already, death according to Buddhism is just a concept. The mind states continue to arise and cease according to the law of causality.

As to how one mind can affect the next, I really have no answer besides that through the practice of meditation you can verify for yourself that it is indeed the nature of reality; why it is this way is somewhat irrelevant.