Use the LINQ Select method.
var newList = oldList.Select(oldItem => new InheritedItem(oldItem)).ToList();
In plain English this translates to "Take each item from oldList, feed each as a function parameter to a function which will take that item and perform some logic to return a different type of item, then take all the returned items and populate them into a new List."
Or if you don't have a constructor to initialize the inherited class then you can provide any code block:
var newList = oldList.Select(oldItem =>
{
var newItem = new InheritedItem();
newItem.Property = oldItem.Property;
return newItem;
}).ToList();
Or an initializer:
var newList = oldList.Select(oldItem => new InheritedItem()
{
Property = oldItem.Property,
Property2 = oldItem.Property2
}).ToList();