  The generic link of an unmixed radical ideal is radical (in fact, prime). We
show that the squarefreeness of the initial ideal and $F$-purity are, however,
not preserved along generic links. On the flip side, for several important
cases in liaison theory, including generic height three Gorenstein ideals and
the maximal minors of a generic matrix, we show that the squarefreeness of the
initial ideal, $F$-purity, and the $F$-pure threshold are each preserved along
generic links by identifying a property of such ideals which propagates along
generic links. We use this property to establish the $F$-regularity of the
generic links of such ideals. Finally, we study the $F$-pure threshold of the
generic residual intersections of a complete intersection ideal and answer a
related question of Kim--Miller--Niu.
