Putting aside why you implement custom mapper, when there are many solutions available, you can try following:

    private static List<TDestination> MapToList<TSource, TDestination>(List<TSource> source)
    {
        var target = new List<TDestination>(source.Count);

        var addMethod = target.GetType().GetMethod("Add");

        foreach (var item in source)
        {          
            TDestination mappedObj = //DOMAP

            addMethod.Invoke(target, new object[] { mappedObj });
        }
        return target;
    }