How to "clone" an object into a subclass object?

You can achieve this by using reflection.

Advantage: Maintainability. No need for changing copy-constructor or similar, adding or removing properties.

Disadvantage: Performance. Reflection is slow. We're still talking milliseconds on average sized classes though.

Here's a reflection-based shallow copy implementation supporting copy-to-subclass, using extension methods:

public static TOut GetShallowCopyByReflection<TOut>(this Object objIn) 
{
    Type inputType = objIn.GetType();
    Type outputType = typeof(TOut);
    if (!outputType.Equals(inputType) && !outputType.IsSubclassOf(inputType)) throw new ArgumentException(String.Format("{0} is not a sublcass of {1}", outputType, inputType));
    PropertyInfo[] properties = inputType.GetProperties(BindingFlags.Public | BindingFlags.NonPublic | BindingFlags.Instance | BindingFlags.FlattenHierarchy);
    FieldInfo[] fields = inputType.GetFields(BindingFlags.Public | BindingFlags.NonPublic | BindingFlags.Instance | BindingFlags.FlattenHierarchy);
    TOut objOut = (TOut)Activator.CreateInstance(typeof(TOut));
    foreach (PropertyInfo property in properties)
    {
        try
        {
            property.SetValue(objOut, property.GetValue(objIn, null), null);
        }
        catch (ArgumentException) { } // For Get-only-properties
    }
    foreach (FieldInfo field in fields)
    {
        field.SetValue(objOut, field.GetValue(objIn));
    }
    return objOut;
}

This method will copy all properties - private and public, as well as all fields. Properties are copied by reference, making it a shallow copy.

Unit tests:

[TestClass]
public class ExtensionTests {
    [TestMethod]
    public void GetShallowCloneByReflection_PropsAndFields()
    {
        var uri = new Uri("http://www.stackoverflow.com");
        var source = new TestClassParent();
        source.SomePublicString = "Pu";
        source.SomePrivateString = "Pr";
        source.SomeInternalString = "I";
        source.SomeIntField = 6;
        source.SomeList = new List<Uri>() { uri };

        var dest = source.GetShallowCopyByReflection<TestClassChild>();
        Assert.AreEqual("Pu", dest.SomePublicString);
        Assert.AreEqual("Pr", dest.SomePrivateString);
        Assert.AreEqual("I", dest.SomeInternalString);
        Assert.AreEqual(6, dest.SomeIntField);
        Assert.AreSame(source.SomeList, dest.SomeList);
        Assert.AreSame(uri, dest.SomeList[0]);            
    }
}

internal class TestClassParent
{
    public String SomePublicString { get; set; }
    internal String SomeInternalString { get; set; }
    internal String SomePrivateString { get; set; }
    public String SomeGetOnlyString { get { return "Get"; } }
    internal List<Uri> SomeList { get; set; }
    internal int SomeIntField;
}

internal class TestClassChild : TestClassParent {}

I would add a copy constructor to A, and then add a new constructor to B that takes an instance of A and passes it to the base's copy constructor.

There is no means of doing this automatically built into the language...

One option is to add a constructor to class B that takes a class A as an argument.

Then you could do:

B newB = new B(myA);

The constructor can just copy the relevant data across as needed, in that case.