How to filter custom field values based on the selection of another field

First of all, let's create a field Country with its values:

and City:

As you can see, by default, all cities are shown independently of the selected country: 

To filter these values, you can use the State-machines per Issue Type workflow rule. The example сode may seem complicated, but do not worry, we will try to simplify it.

For this, create a separate custom script and call it, for example, transitions-helper:

var GenerateTransitions = function(TypeValuesMapping, TypeValue) {
  var targers = {};
  TypeValuesMapping[TypeValue].forEach(function(value) {
    var target = {
      targetState: value
    };
    targers[value] = target;
  });
  return targers;
};

var GenerateStateMachine = function(TypeValuesMapping, TypeValue) {
  var requiredValues = TypeValuesMapping[TypeValue];
  var stateMachine = {};
  var needSetInitial = true;
  requiredValues.forEach(function(elem) {
    var item = {};
    item.transitions = GenerateTransitions(TypeValuesMapping, TypeValue);
    if (needSetInitial) {
      item.initial = true;
      needSetInitial = false;
    }
    stateMachine[elem] = item;
  });
  return stateMachine;
};
var GenerateAlternativeMachines = function(TypeValuesMapping) {
  var alternativeMachines = {};
  for (var typeValue in TypeValuesMapping) {
    alternativeMachines[typeValue] = GenerateStateMachine(TypeValuesMapping, typeValue);
  }
  return alternativeMachines;
};
exports.GenerateStateMachine = GenerateStateMachine;
exports.GenerateAlternativeMachines = GenerateAlternativeMachines;

Then, create a separate State-machines per Issue Type rule and add a reference for transitions-helper:

var helper =  require('./transitions-helper');

Create a list of cities that should show for each country. Please note that all values should be the same as values in fields Country and City: 

var cities = {
  'Czech Republic': ['Prague'],
  Netherlands: ['Amsterdam', 'Rotterdam'],
  Germany: ['Berlin', 'Hamburg', 'Munich'],
  Russia: ['Moscow', 'Saint Petersburg'],
  USA: ['New York', 'Los Angeles', 'Chicago']
};

Then, specify the stateFieldName and typeFieldName properties:

stateFieldName: 'City',
typeFieldName: 'Country',

Now, set the default state machine using the GenerateStateMachine method from our helper:

defaultMachine: helper.GenerateStateMachine(cities, 'Russia'),

And finally, add alternative state machines:

alternativeMachines: helper.GenerateAlternativeMachines(cities),

Here is the complete code:

var entities = require('@jetbrains/youtrack-scripting-api/entities');
var helper =  require('./transitions-helper');

var cities = {
  'Czech Republic': ['Prague'],
  Netherlands: ['Amsterdam', 'Rotterdam'],
  Germany: ['Berlin', 'Hamburg', 'Munich'],
  Russia: ['Moscow', 'Saint Petersburg'],
  USA: ['New York', 'Los Angeles', 'Chicago']
};

exports.rule = entities.Issue.stateMachine({
  title: 'Cities per Country',
  stateFieldName: 'City',
  typeFieldName: 'Country',
  defaultMachine: helper.GenerateStateMachine(cities, 'Russia'),
  alternativeMachines: helper.GenerateAlternativeMachines(cities),
  requirements: {}
});

I hope it will be helpful!