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Here is my code that is not working.

vars = <|"x" -> <|"Q" -> "Quant", "bins" -> {10, 20, 30, 40}|>,
"y" -> <|"Q" -> "Qual", "bins" -> {"Male", "Female"}|>|>;

doVarSpecs[vars_] := DialogInput[
DynamicModule[{varlist, var, oplistquant, oplist, op, vallist, val},
g[var_] := Block[{},
 oplist = 
  If[vars[var, "Q"] === "Quant", Keys@oplistquant, 
   Take[Keys@oplistquant, 2]];
 op = First@oplist;
 Print[op, oplist];
 vallist = vars[var, "bins"]; val = First@vallist;
 ];
varlist = Keys@vars; var = First@varlist;
 oplistquant = {"=" -> SameQ, "\[NotEqual]" -> UnsameQ, 
 "\[LessEqual]" -> LessEqual, "\[GreaterEqual]" -> GreaterEqual};
 g[var]; Print[varlist]; Print[oplist]; Print[vallist];
 Column[{
  Style["Choose a Set of Variable Relations", 22, Darker@Blue],
  "  ",
  Row[{
   "Variable: ",
   PopupMenu[Dynamic[var, (var = #; g[var]) &], varlist], "  ",
   Dynamic[PopupMenu[Dynamic[op], oplist], 
    TrackedSymbols :> {oplist}], "  ",
   Dynamic[PopupMenu[Dynamic[val], vallist], 
    TrackedSymbols :> {oplist}]
   }],
  "  ",
  Row[{Button["OK", DialogReturn[{var, op, val}]], CancelButton[]}]
 }
 ]
], WindowTitle -> "Variable Specification Selector"]

For a simpler case, the question was answered previously,

My previous similar question

In this case, when var is changed, g[var] is called and both oplist and vallist are updated, but the PopupMenu controls that use them are not updated.

The assistance of the Stack Exchange community would be greatly appreciated.

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I found a mechanism that works for my needs in this case. The key seems to be embedding pure functions (making all of the symbols visible to Dynamic). A code fragment follows:

doVarSpecs[vars_] := DialogInput[
 DynamicModule[{varlist, var, oplist, oplist1, oplist2, op, val},
 varlist = Keys@vars; var = First@varlist;
 oplist = <|"=" -> SameQ, "\[NotEqual]" -> UnsameQ, 
  "\[LessEqual]" -> LessEqual, "\[GreaterEqual]" -> GreaterEqual|>;
 oplist1 = Keys@oplist; oplist2 = Take[oplist1, 2];
 q = vars[var, "Q"] === "Quant";
 Column[{
  Style["Choose a Set of Variable Relations", 22, Darker@Blue],
  "  ",
  Dynamic@Row[{
    "Variable: ",
    PopupMenu[
     Dynamic[var, (var = #; q = (vars[var, "Q"] === "Quant"); 
        op = First@oplist1; val = First@vars[var, "bins"]) &], 
     varlist, MenuStyle -> 18],
     "  ",
    PaneSelector[{
      True -> PopupMenu[Dynamic[op], oplist1, MenuStyle -> 20],
      False -> PopupMenu[Dynamic[op], oplist2, MenuStyle -> 20]
      }, q], "  ",
    Dynamic@
     PopupMenu[Dynamic[val], vars[var, "bins"], MenuStyle -> 18]
    }],
   "  ",
   Row[{Button[" OK ", DialogReturn[expr], BaseStyle -> 18], "  ",
    Button[" Cancel ", DialogReturn[{}], BaseStyle -> 18]}]
   }, BaseStyle -> 18]
  ], WindowTitle -> "Variable Specification Selector" ]

Example call

vars = <|"Age" -> <|"Q" -> "Quant", 
    "bins" -> {10, 20, 30, 40, 50, 60, 70, 80}|>,
 "Gender" -> <|"Q" -> "Qual", "bins" -> {"Male", "Female"}|>,
 "Race" -> <|"Q" -> "Qual", 
    "bins" -> {"White", "Black", "Hispanic", "Asian", "Other"}|> |>

doVarSpecs[vars]
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