So the concept stewardship is usually invoked in a context where something valuable is being shared over time. In ethical terms that suggests that sharing over time is good; that is to say, it's a virtue, a duty, or a preferable consequence.
If stewardship is a virtue it signifies excellence in character. For Aristotle, it would relate to distributive justice. Hence, a "good person" takes no more nor less than he/she deserves. Now Aristotle's Nicomachean Ethics really doesn't get into intergenerational justice, but we know from his other writings that potential beings would have moral standing. We also know that if stewardship is a virtue, it would involve teaching, learning, and the establishment of a habit. But beyond that I'm not sure Aristotle takes us very far. The Judeo-Christian concept of the virtue of stewardship is based on the idea that God gave the universe to humans to share. There is a built in sense of value associated with any "gift" that comes from an omnipotent, omniscient, omnipresent, and "Good" being. God, by definition, does not give lousy gifts! No ugly ties or exploding cigars! Moreover, since God created Homo sapiens as an intergenerational community, he certainly would not favor early generations over subsequent ones. Nature is not a Ponzi scheme! Hence, previous generations took their fair share, we take, our fair share, and future generations take their fair share. The problem here is how do we know how much each generation can consume without shortchanging the next generation? The history of humans on earth suggests that virtue-based concepts of stewardship have led to intergenerational exploitation rather than intergenerational justice.
If stewardship is a duty, then we must think about it in a different way. First of all, we'd have to establish that all generations have an equal right to that good thing that is being shared between generations. So if we establish that each generation has an equal right to the fish in ocean then, each generation has a duty to determine it's fair share of fish and preserve the rest for the next generation. But there's a lot more here than meets the eye. Suppose subsequent generations will have more living persons than our generation? Do we have a duty to take into account the fact that the next generation will "need" more than we do? And, of course how far into the future does this obligation extend? Moreover, the human ability to exploit the earth's bounty changes relative to technology. We can catch more fish, cut down more trees, extract more oil and coal than previous generations. Hence, the potential for intergenerational injustice is magnified over time. All I want to say here is that even if we all agree that we have a duty to hold back a few fish for future generations we really don't know what this duty entails. In short, it's a lot easier to assign rights and duties across generations than it is to fulfill them.
Finally, if stewardship is conceived as the means to a preferable set of consequences or outcomes across generations, we must be able to assess cost/benefit ratios across generations. Thus stewardship might imply aiming at the "greatest happiness for the greatest numbers" across generations. If this is our goal, then it's not at all clear how much we ought to hold back, given that we really don't know how many generations will follow us. I would argue that if stewardship requires utilitarian calculus exercised across generations, each generation's rate of consumption would be minicule if not zero. Of course, that would solve the obesity problem, but that's another topic worth exploring.
So what does all of this say about stewardship as a moral concept? First of all, there's a lot of muddled thinking about "sustainability." If we have a moral obligation to consume at a "sustainable" rate we have to decide how far into the future that obligation extends. If we limit our obligation to the next generation we might be able to calculate a sustainable rate of consumption. On the other hand, if todays politicians seek to be responsible stewards of the earth's bounty and neglect the needs (and wants) of our present generation in order to save a few fish for future generations, those politicians won't be in office very long. Future generations aren't old enough to vote!