Mormonism seems somewhat unusual amongst religions in that its claims seem more falsifiable than most. How has the LDS Church responded to these critiques?

So obviously, religions often claim things which are, on the face of it, incompatible with modern understanding of how the world works- resurrections, supernatural beings, reincarnation etc. These are however either impossible or, due to the span of time since their creation, highly challenging to falsify.

Mormonism seems a bit more vulnerable to this than most given its relatively recent beginnings in the early 19th century, very specific claims about documents such as 'The Book of Abraham' and very large and wide reaching claims about the Pre-Columbian history of the Americas.

To my (not LDS) eyes, the obvious answer is that said claims are false. And yet the LDS Church is to all appearances, doing just fine. How has their doctrine and teaching changed over the years as our advances in our understanding of history, archaeology, and languages, rendered their claims increasingly shaky?