A typical LVDT output is below. As you can see, outside the range the output falls off.

A reason I have seen given for this is that it recieves less flux from the primary, as per this next picture

However is it not also true that for a transformer, the output determined by the ratio of primary to secondary turns, now in this case, the primary turns are decreasing so should the output not increase?

I've clearly got something wrong, so clarification would be wonderful!

 

The primary turns is the same but with the core partially withdrawn you are only using part of the magnetic field produced by the primary.

 

Think about it: if you remove the core completely, both secondaries couple equally to the primary -- as if the core were in the center position. It stands to reason that, as the core is withdrawn, at some point the rate of rise of amplitude vs. distance will drop and, ultimately, reverse until the output drops to zero.