Ken's answer is pointing you in the right general direction. However, very likely the decisions will be made well above the actual school. It's sometimes easier if it's within the same city or county. There generally will be some sort of procedure (or procedures) that the school board has established on when it will permit a child who lives within the boundaries of one school to attend another school.
The ease with which this can be done will vary, based on the city or county. Also, very frankly, it can vary depending on the racial or ethnic makeup, or other factors, of the two areas. For instance, the school board may want greater diversity in a school. If you're trying to transfer a child out of the school, and the child belongs to the group the system wishes to retain or attract, it may be very difficult. On the other hand, if your child belongs to a group that the new (that is, your hoped for) school wishes to attract, you may find it very easy to make the transition. You may also find anomalies if the tax rates in the two jurisdictions are dissimilar. For instance, if you're in a district that taxes homeowners (or allocates for schools) $3,000 per student, and you're trying to transfer to a district that spends $8,000 per student, you may find the new jurisdiction reluctant to accept your child.
Hope that helps.