While I believe it would be possible to make a falling block action which could shoot with the bolt actions, I also think it would be difficult to do. Understand, I am speaking of BR level accuracy. There is little question that falling blocks can shoot well enough that there is little difference in the hunting fields.
The successful single shot would probably have an inline striker. If a two-piece stock design was retained, the fore arm would be attached to a rigid hanger and the buttstock well fitted and attached. Scope mounting would be on a cantelevered extension integral with the receiver. The requirements vis a vis squareness of the breechface etc. would be just as strigent as with a precision bolt action.
The bolt action is the easy way to achieve precision at this time.
As I said before, the falling block is certainly capable of excellent field accuracy. I have owned and built No. 1's which were true 1/2 moa rifles and I really like them but I don't see a falling block target rifle in my future. GD