You can set the object's velocity immediately after instantiating it.

In general, you shouldn't use the ability to set velocity directly as a method of controlling or moving the object (you should apply forces instead), but in this situation where you are setting its initial velocity after creating it, it should be fine:

// create the new object
var newObject = Instantiate( sourceObject, position, rotation );

// give it the same velocity as the current object
newObject.rigidbody.velocity = rigidbody.velocity;
newObject.angularVelocity = rigidbody.angularVelocity;

// destroy this object
Destroy(gameObject);

You might also find that you need to set up an IgnoreCollision between the old and new objects - if they both have colliders, there's a chance a physics update may occur while they both exist in the same position.