Some results and directions in finite model theory

I will discuss the FO-transduction order on classes of graphs (or other structures), defined by the relation: a class C can be obtained by an FO-transduction from a class D. I will focus on the interplay of this order with notions such as treewidth, clique-width, twin-width, bounded expansion, nowhere denseness, monadic stability, monadic NIP. I will state some known results and pose several conjectures. For example, one of the conjectures is a strengthening of a conjecture of Seese regarding MSO transductions.