This sentence is complex, and made more complex by the way that it has been punctuated. Simply altering the punctuation can clarify some of the meaning:
And it shall come to pass, (when the Lord seeth fit in his wisdom) that they shall minister unto all the scattered tribes of Israel, and unto all nations, kindreds, tongues and people. And shall bring out of them unto Jesus many souls, that their desire may be fulfilled. And also because of the convincing power of God which is in them.
The remaining difficult clause is “that their desire may be fulfilled.” That phrase could conceivably relate to the desire of those who are brought out of the many nations. It could also refer to the desire of the three. Since the next phrase that begins with “and also” certainly refers to the three, it makes most sense for the “their desire” would also pertain to the three. However, Mormon has not clearly given us the nature of that “desire” in his description. We are left to assume that their desire is to bring souls to the Lord, which would certainly be in character for apostles of the Lord, but is not explicitly stated in Mormon’s relation.