As to the specific issue of cultivating compassion for others, the Buddhist view seems predicated on the metaphysical notions of cyclic existence and of the lack of inherent existence of things, including the self, in the world. As Christians, we get to the same result, the urgency and necessity of compassion, without the baggage of cyclic existence and without needing to view the self or objects in the world as somehow illusory (although we do agree on the transiency of what we see around us).
For the Christian, the ups and downs of cyclic existence (reincarnation) are the ups and downs of our one life on earth. From a Christian point of view, the idea of reincarnation is a purely symbolic, non-realistic fable or story that communicates a truth about this one life on earth: our good and evil actions have real consequences. The Dalai Lama focuses on the need to cultivate compassion because we have gained the insight that all of us are caught in the same web of suffering and that all of us desire happiness.
The Christian approach to the same conclusion (compassion and solidarity with others) lies in the very character of the God who created and sustains the universe: the deity is love. As such, the law of our own being as creatures of that deity is also love. Our true identity is to be gift to the other. In being gift to others, we find our authentic worth, mission, and happiness. We find our escape from our own suffering. The Christian metaphysical underpinning of compassion does not depend on believing in reincarnation and does not depend on emphasizing the illusory nature of existence. Rather, the Christian underpinning is based on the law of love inscribed in our very being in a world whose engine is love. I highly recommend that Christians take a look at how Buddhism celebrates compassion and see how the Christian tradition in a different way comes to the same conclusion: the key to our existence is cultivating compassion for others.
Ultimately, the question we need to ask every day is this: am I--in what I habitually talk about and in what I habitually do--being "gift" to others or a burden? Being gift is to be Christ, being a burden is to be an Antichrist. We do not need any more Antichrists.