When we talk about no separate self we have to actually put it more into Buddhist terms because there's a lot of confusion about this. Talk about "Anatta." Some people when they hear the word Anatta or "no-self" they start to develop the idea that there is nothing, but this is not what we understand Anatta means in Buddhism. An analogy is: Suppose you went to the marketplace, and you wanted some juicy apples, but the seller shows you some mandarins, and you say those are not apples. Just because we're saying those are not apples doesn't mean the mandarins don't exist, it just means that the mandarin is not an apple. So we prefer to translate the third characteristic as "not-self" rather than no-self because no starts to give a lot of difficult and possibly wrong interpretations of Anatta. And it can lead many people into believing that there is nothing.
When we investigate Anatta in Buddhism, we're more investigating to see if there is anything permanent, rather than if there is nothing. The "I" that we consider ourselves to be depends on other impermanent things. Usually when people think of "I," they have a fixed ideal of themselves, that "I" am this and this and this. So when we investigate into what we consider ourselves to be, we see that all these things are impermanent and dependently arising. Therefore this "I" is constantly changing, it is not a fixed idea.
Another analogy I like to use about this is considering a river. Usually when people consider a river, say the Mississippi River, they have an idea, well Mississippi River exists, it's right there on a map, it's in the United States, right? But if you go to the side of the Mississippi river, and you look at one spot in the middle of the river, is it the same water going by? Usually we see it's not the same water at all, it's constantly changing, but the concept, Mississippi River, remains the same. And so many people get attached to this "I" concept, the "I am this I am that," not seeing that the "I" that we are, or think we are, is constantly changing, and dependent on other conditioned things that arise and pass away. So this is more what we understand it means about not-self, Anatta, that there's no permanent fixed entity, that everything that we consider ourselves to be is constantly arising and passing and dependent on other impermanent things.
This is quite in line with understanding Kamma, cause and effect, because when we look at Kamma it is more a process of arising and passing, arising and passing depending on conditions. So Anatta is not in conflict with this. If we're serious about wanting to know whether there is an ultimate self or no ultimate self, it's important to investigate into what we are not. Because most of us are terribly attached to what we are not: this body, this mind.
So investigate into it and start to let go of attachment to what we are not, and perhaps then we'll come to an understanding of what we really are. Taking that analogy of the Mississippi river again, what goes into the "next life" is still the Mississippi river, it's just a flowing river, right? It continually keeps flowing depending on cause and effect. From causes which have been laid, comes the results. We can also understand this more by considering ourselves when we were 6-years-old, examining the body and mind of a six year old, and comparing it to who we are now. This body that I am now, is not the 6 year old body, but is dependent on the 6 year old body to be here now. But it is certainly not the same body. It's been constantly changing, constantly changing. And the mind, and all the thoughts and perceptions and mental formations and everything that occupied the 6 year old don't usually occupy my mind now. Yet a lot of the thoughts that arise may have been dependent on the 6 year old. So usually we have an idea, a concept, me, and "I," Rosemary, and all our "statistics." And we think it's something solid, but it's just statistics, like the Mississippi River, describing more this flow of this body and mind that is arising due to causes and conditions.
That's why we have to have compassion for the person we will become in the future. We might be quite different to who we are now. Because some of the results that we receive in the present, sometimes we don't think it's fair because we've changed! We've changed a lot, so "Why do I have to get all the results of that! I don't believe all those things that I believed when I was little! You know I don't do that anymore, why should I get all the results?!" We may have increased our understanding, but the law of cause and effect continues. What you sow, you shall reap. Cause and effect. So we have to try to have compassion for the person we may become in the future, because that person will have to reap the results of what we do now. As far as future lives, it seems obvious to me, at least it seems intelligent to consider, if there's a next life, the laws of nature won't suddenly change, and we want to protect ourselves if there is a next life and do beneficial actions in this life.