How in buddhistic terms one can explain life evolution? How can in Buddhistic terms explain life evolution, that is how can new beings evolve  from previous nama-rupa? How could a vipāka (resultant-consciousness) "move" from cuti-citta (dying-consciousness) to the patisandhi-citta (rebirth-consciousness) of a being not yet existing?
Here is a quote from the book "A Comprehensive Manual of Abhidhamma" by Ven. Bhikkhu Bodhi. The quote is from the chapter: "The Process of Death and Rebirth", p. 220-223:


  §36 The Mind at the Time of Death
  
  Thereafter, attending to that object thus presented, the stream of consciousness in accordance with the kamma that is to be matured, whether pure or corrupted, and in conformity with the state into which one is to be reborn continually flows, inclining mostly towards that state. Or that rebirth productive kamma presents itself to a sense door in the way of renewing.
  
  Guide to §36
  
  In the way of renewing (abhinavakaraoavasena): that is, the kamma presenting itself does not appear as a memory image of something that was previously done, but it appears to the mind door as if it were being done at that very moment.
  
  §37 Death and Rebirth-Linking
  
  To one who is on the verge of death, either at the end of a cogni- tive process or at the dissolution of the life-continuum, the death con- sciousness, the consummation of the present life, arises and ceases in the way of death.
  Immediately after that (death consciousness) has ceased, a rebirth-linking consciousness arises and is established in the subsequent existence, apprehending the object thus obtained, either supported by the heart-base or baseless, as is appropriate; it is generated by a volitional formation that is enveloped by latent ignorance and rooted in latent craving. That rebirth-linking consciousness, so called because it links together the two consecutive existences, is conjoined with its mental adjuncts, and acts as the forerunner to the conascent states as their locus (or foundation).
  
  Guide to §37
  
  To one who is on the verge of death: The last cognitive process begins when the bhavanga is interrupted, vibrates for one moment, and is then arrested. Thereafter follows either a sense-door process taking as object some sense object presenting itself at one of the five sense doors or a bare mind-door process taking as object either some sense object or a mental object presenting itself at the mind door. Within this termi- nal process the javana phase, by reason of its weakness, runs for only five mind-moments rather than the usual seven. 
  
  This process lacks original productive kammic potency, but acts rather as the channel for the past kamma that has assumed the rebirth-generative function. Following the javana stage two registration cittas (tad±rammaoa) may or may not fol- low. In some cases the bhavanga may follow the last process cittas. Then, as the very last citta, the death consciousness arises performing the func- tion of passing away from the present life. With the ceasing of the death consciousness, the life faculty is cut off. Then the body remains a mass of inanimate material phenomena born of temperature, and continues as such until the corpse is reduced to dust.
  
  Immediately after that has ceased: Following the dissolution mo- ment of the death consciousness, there arises in a new existence the re- birth-linking consciousness apprehending the object thus obtained in the final javana process of the previous life. This citta is supported by the heart-base in realms which include matter, but is baseless in the imma- terial realms. It is generated by a volitional formation, i.e. the kamma of the previous javana process, which in turn is grounded in the twin roots of the round of existence, latent ignorance and latent craving. The rebirth consciousness is conjoined with its mental adjuncts, i.e. the cetasikas, which it serves as a forerunner not in the sense that it pre- cedes them, but in that it acts as their locus (or foundation).