Reading the documentation for the Azure Search .NET SDK, I see that the ContinuationToken property is not supposed to used for pagination (this is the same as the @odata.nextLink and @search.nextPageParameter properties in the REST API).
Note that this property is not meant to help you implement paging of search results. You can implement paging using the Top and Skip search parameters. Source
Why can't I use it for pagination? I have a situation where I want to run a query and then step through a static copy of the results page by page. I don't want those query results to change beneath my feet, however, as I am navigating through them, as new documents are added to the underlying database. In my case, there could be hundreds or thousands of results that get added in the minute or two between submitting the initial query and navigating to another page. How could I accomplish this?