Visibility issues with validating FormFields' fields

Quick Description

field.pre_validate() runs for normal fields as well for fields inside a FormField, displaying all errors as expected.
However, field.process_formdata() does not seem to run for fields inside a FormField, or the errors from it are not collected.
There also seems to be some issues with setting data to None outside the FormField.
This leads to inconsistent behaviour - we only see some errors, and some of the problematic data has been set to None.

Setup

>>> from flask import Flask
... from flask_wtf import FlaskForm
... from werkzeug.datastructures import ImmutableMultiDict
... from wtforms import fields
... 
... app = Flask(__name__)
... app.config["WTF_CSRF_ENABLED"] = False
... 
... class FF(FlaskForm):
...     choices_inner = fields.SelectField(label="Test Select", choices=(("1",11), ("2",22)))
...     date_inner = fields.DateField()
... 
... class TestForm(FlaskForm):
...     choices = fields.SelectField(label="Test Select", choices=(("1",11), ("2",22)))
...     date = fields.DateField()
...     form = fields.FormField(FF)
...         
... with app.app_context():
...     test = TestForm(formdata=ImmutableMultiDict({"choices": 3, "date": "20000", "form": {"choices_inner": "3", "date_inner": "20000"}}))

Actual Behavior

>>> test.validate()
False
>>> test.errors
{'choices': ['Not a valid choice'], 'date': ['Not a valid date value'], 'form': {'choices_inner': ['Not a valid choice']}}
>>> test.data
{'choices': '3', 'date': None, 'form': {'choices_inner': None, 'date_inner': None}}

Expected Behavior

>>> test.validate()
False
>>> test.errors
{'choices': ['Not a valid choice'], 'date': ['Not a valid date value'], 'form': {'choices_inner': ['Not a valid choice'], 'date_inner': ['Not a valid date value']}}
>>> test.data
{'choices': None, 'date': None, 'form': {'choices_inner': None, 'date_inner': None}}

Environment

• Python version: 3.7.3
• wtforms version: 2.3.1

Start with the provided Flask/WTForms reproducer and trace FormField validation, especially pre_validate() and process_formdata(). Confirm the fix by checking that nested date errors are collected and that invalid field data matches the expected None values in test.errors and test.data.