Suggested answer:

(1) Of the four, besides and the string what is more seem to have other functions than conjunct, as in the examples below:

E.g. besides as a preposition with the meaning of apart from / other than:

The yellow chair, besides being brushstrokes and pigment, besides being a yellow chair, was one of twelve...

What is more does not alway function as a conjunc unit. In the following example the words in the string function as separate constituents in the clause:

What is more important from my point of view is that they can often ...

(2) The corpus results show that when both moreover and what is more are in first position they are interchangeable, e.g.:

Moreover, they have seen that it is possible to break out of ...
What is more, they posed no great problems for his rather generalised...

They do not seem to be interchangeable in any other position, e.g. midposition:

Foreign places, moreover, bring out the writer in strangers less word-obsessed...

In this and similar contexts, what is more is ruled out.

(3) In general, a conjunct relates what is said in a sentence to another sentence. The four conjuncts under discussion here are all of the additive type; they serve to add on to what was said in the previous sentence.

Read about conjuncts here and here (Power Point file (see slides 9-12))
Terminology: conjunct