  In a previous paper [22] the author studied the directed weak covering
homotopy property (dWCHP)and directed weak fibrations in the category dTop of
directed spaces in the sense of M. Grandis [12], [13], [14]. This type of maps
extend to the category dTop the well known Dold's (or weak)fibrations [6]. In
this paper the transition from local to global of the dFHE (directed fiber
homotopy equivalence) and the dWCHP are studied by proving two Dold type
theorems and respectively a tom Dieck-Kamps-Puppe type theorem [3]. Some new
notions of directed topology are defined: d-halo, d-SEP, d-numerable covering,
d-shrinkable. Some examples and counterexamples are given.
