After the homestudy is completed you will wait for the assignment of a child. Families who are the most flexible tend to wait the shortest amount of time. Those who are asking for children with special needs move quickly. With the assignment made, you must file all the necessary documents for Immigration and Naturalization Services. Then you wait again until the child is cleared to travel.
Some countries require that the family travel to that country for the adoption.
A post-placement study is done according to the requirements of your state. After the adoption is finalized your child may become an American citizen. It depends on the agency and the country. Costs can vary greatly, again depending on the circumstances. There are some programs that may help facilitate special needs adoption, but one would have to ask the agency about those programs.