To answer your question, Yes, it is possible to transfer dental schools.
However, this is not common and there are numerous requirements and limitations for those schools that have a transfer policy.
First of all, there would have to be a seat available for a transfer to even occur. Most dental schools operate at capacity. If the class size is 80 students, the school can support exactly 80 students. Every school has the occasional classmate that drops out for one reason or another, or repeats a given grade. But this does not tend produce any significant number of 'open' seats.
Most programs would only allow a transfer before the clinical phase begins - this would mean that a transfer would typically occur within a narrow time window - usually at the beginning of second year.
The school of origin would have to have a very similar curriculum structure. Principally, some schools have a fairly traditional - first two years didactic, last two years clinical structure; and other schools feature an integrated curriculum, offering clinical exposure early on in the education process.
Transfer applications do require a separate, official transfer application. I cannot speak to what the application entails. As to the cost and competitiveness, I do not believe there is any released data to support the process as more rigorous or expensive.
I would imagine if you can 'get in' to one, you could do the same at another. Though I'm sure any grades and records earned while at the dental school of origin would constitute the defining qualification factor.
One school that comes to mind immediately would be Columbia. They have a policy outlined online, but most of the pertinent information would require direct contact.
Hope that helps.