|Lotus, symbol of karma.
The idea that we live on in the hearts and minds of those we influence appears in many religions and philosophies. When this "afterlife" is combined with the persistence of the consciousness of the spirit, it typically takes the form of reincarnation, in which the spirit finds itself in yet another body, ready to repeat the life cycle. The generic term for the spirit finding a new host is "transmigration", while some systems attach additional significance to this, such as reincarnation in Hinduism, wherein the spirit is required to live as many times as necessary before learning all of the lessons, and achieving Nirvana.
EBS considers this to be artificially limiting. First, our spirits do not wait until we die to spread into other hosts — whenever we influence someone, our spirits join the community of spirits already in the hosts. Second, it isn't that one body hosts one spirit which then needs to get another body allocated to it on the death of the former host — one spirit can spread into many different hosts. So the only thing that happens at the death of the body is that the spirit inside it can no longer spread into other hosts, because it can no longer initiate new influences on others. Thus the transmigration doesn't happen at death — it happens during life, and ceases at death, whereupon only those transmigrations that have already occurred can sustain the spirit.
Thus the EBS conception of spiritual endurance, despite the transience of biological hosts, is closer to the concept of karma than it is to the concept of reincarnation.