Updating linked server with temp table join
current query  

Update linked 
   set linked.col1 = local.col1
  From LinkedServer.LinkedDatabase.dbo.remoteTable linked
  JOIN #localTempTable local 
         on local.column = linked.primaryKeyColumn 


It is typically better to perform the write operation locally
Here you are both reading from and writing to the linked server   

My recommendation is if you are going to create #localTempTable then do so on LinkedServer.  Have a session on the LinkedServer and link to local to populate #localTempTable but #localTempTable is now on LinkedServer.  Flop the link.   

insert into #linkedLocalTempTable 
select ... 
from Local.LinkedDatabase...


Please post the query to populate #localTempTable
If you use that table for other stuff then flopping stuff around may not be and option