I would suggest not taking them away from the page unless there is good reason too. A better way would be to manipulate the DOM (i.e update a container with the message) or display a dialog with the message (usually my choice unless they are loading the page).
Maybe walk us through step by step as a member of how it would work.
- Member clicks link in navigation to go to adopt a pet page.
- Page shows all possible pets the member can adopt.
- Member clicks on pet they wish to adopt.
- Dialog pops up asking them to confirm.
- If confirmed, the pets container is hidden OR a dialog pops up, and the congratulations message is displayed.