Buddhism on the other hand says that the person is made up of thoughts, feelings and perceptions interacting with the body in a dynamic and constantly changing way. At death this stream of mental energy is re-established in a new body.

Critics of the Buddhist doctrine of rebirth say that if there is no soul, only a changing stream of mental energy, then there could be no identity and thus to talk of a person being reborn or experiencing the results of good or bad actions done in the past, is meaningless. However this criticism fails to understand the phenomenon of identity in change. Even within a single life we can notice a person change, sometimes quite dramatically, and yet still be able to recognise them as the same person. This is possible because different aspects of the person changes at different velocities. For example, the complexion and amount of wrinkles on a person's face may change with age while the general shape of the face changes little. Again, a person may change their beliefs while holding them with the same intensity as they held their former ones or perhaps retain the same beliefs but in a more moderate way than before. To use a simile - the Ganges River is changing every moment and over the centuries its width, its course, the quantity and quality of the water it contains have all changed and yet it can still be recognised as the same river. Thus the idea of a dynamic personality does not contradict the idea of identity.

http://www.buddhanet.net/e-learning/dharmadata/fdd47.htm

what we are is determined very much by how we have thought and acted in the past. Likewise, how we think and act now will influence how we will be in the future. The gentle, loving type of person tends to be reborn in a heaven realm or as a human being who has a predominance of pleasant experiences. the anxious, worried or extremely cruel type of person tends to be reborn in a hell realm or as a human being who has a predominance of painful experiences. The person who develops obsessive craving, fierce longings, and burning ambitions that can never be satisfied tends to be reborn as a hungry ghost or as a human being frustrated by longing and wanting. Whatever mental habits are strongly developed in this life will continue in the next life. Most people, however, are reborn as human beings.

http://www.buddhanet.net/e-learning/qanda05.htm

So if we believe this we can see that the qualities we posses in this life as we follow the path continue into our next life.