It looks like the container of daughters is an anonymous array, so given your 'original data' and 'desired re-ordered data' it all amounts to re-ordering the daughters array by their name. Maybe you want a method 'walk_down_sorted' or monkey-patch the module in walk_down amounting to:

--- /usr/share/perl5/Tree/DAG_Node.pm    2018-02-13 23:16:43.000000000
+ +0100
+++ /home/shmem/perl/lib/Tree/DAG_Node.pm    2018-04-23 03:46:29.60751
+5296 +0200
@@ -1661,7 +1661,7 @@
     if(@daughters) {
       $o->{'_depth'} += 1;
       #print "Depth " , $o->{'_depth'}, "\n";
-      foreach my $one (@daughters) {
+      foreach my $one (sort {$a->name cmp $b->name} @daughters) {
         $one->walk_down($o) if UNIVERSAL::can($one, 'is_node');
         # and if it can do "is_node", it should provide a walk_down!
       }
[download]

result with that:

 root    HASH (0           ): { nc => 0, top_foo => 1 }
    a    HASH (0:0         ): { nc => 1 }
    b    HASH (0:0:0       ): { nc => 2 }
    c    HASH (0:0:2       ): { nc => 6 }
    d    HASH (0:0:2:0     ): { nc => 7 }
    e    HASH (0:0:1       ): { nc => 3 }
    f    HASH (0:0:1:0     ): { nc => 4 }
    g    HASH (0:0:1:0:0   ): { nc => 5 }
    h    HASH (0:1         ): { nc => 8 }
    i    HASH (0:1:0       ): { nc => 9 }
    j    HASH (0:1:0:0     ): { nc => 10 }
[download]

A sort callback for sorting daughters in walk_down would be nice to have. File an enhancement request ;-)

edit: this sorting callback should also be available while populating the 'daughters' anonymous array.