How to mock An Abstract Base Class
From this answer it looks like what you need is something along these lines:
[Test]
public void MoqTest()
{
var mock = new Moq.Mock<AbstractBaseClass>();
// set the behavior of mocked methods
mock.Setup(abs => abs.Foo()).Returns(5);
// getting an instance of the class
var abstractBaseClass = mock.Object;
// Asseting it actually works :)
Assert.AreEqual(5, abstractBaseClass.Foo());
}