Slow Linked Server INSERTs on SQL Server 2014
"My local and remote servers were all using SQL Server 2008 R2 but the
  local ones were recently upgraded to SQL Server 2014 and the
  performance of Remote Queries has been crippled (literally from
  minutes to hours)."


When Sql Server performance degrades substantially, my first response is usually "what has changed?"

Well - you just upgraded the local server to Sql Server 2014 which introduces a new Cardinality Estimator (CE).  

As quoted in the blog (Careful Testing the 2014 Cardinality Estimator with Trace Flags):


  When you migrate to SQL Server 2014, rather than turning on the new
  cardinality estimator right away, it's recommended to stick with the old
  cardinality estimator initially by leaving your database in SQL Server
  2012 compatibility mode. Let the database settle down for a few weeks,
  let folks air out their grievances, and give yourself some time to
  make sure the system is working at least as well as the old system.


You say that "the DBA (for the local server) claims that the database configuration is consistent with the old setup".  So, how can you tell if you are being impacted by the new CE?  There are a few ways.

First, 
run this query against the local instance.

select name, compatibility_level from sys.databases


If the compatibility level is 120 or higher, you certainly 'might' be using the new CE (assuming it has not been overridden by using session or global trace flags, or QUERYTRACEON hints on the query that force the use of the older CE.)

Another way to see if the new CE is being used is to look at the execution plan XML for a 'local' query and locate the string 'CardinalityEstimationModelVersion'.



If it's 70, you're using the old CE.  If it's 120, you're using the new CE.

My suggestion is to 'first' figure out whether you are being bitten by the new Sql Server 2014 CE.

There is quite a bit of information on the Internet about this topic. 

Another helpful link is The SQL 2014 Cardinality Estimator Eats Bad TSQL for Breakfast.