SQL Server – Query to Get Yesterday’s Data for Specific Time Interval

Well I have some ideas which could cause this behavior.

1. Do you monitor your memory pressure? Maybe your queries raise a certain limit which will cause the flush of the plan cache. I don't know your application, but does this correspondent with your logs from your frontend servers? Is there pressure too during this time?
2. Do you have a dedicated SQL Server or does the server share it's hardware with other processes/services? If not, try to consider to outsource your SQL Server to a dedicated machine instead. This will reduce side effects from other services.
3. You may want to use optimize for ad hoc workloads, which will just save a plan stub and compile it if it's needed. This will reduce the load of your plancache, which will lower the chance of an plancache flushing. You can enable it using sp_configure 'optimize for ad hoc workloads',1; reconfigure. This can be done if you have enabled the advanced options using sp_configure 'show advanced options',1; reconfigure.
4. Another idea can be backups. Just simple backups. If they are aggressively, it may occur that your machine gets under pressure too. The time your mentioning just sounds like a good timespan for planning a backup.
5. Maybe it's quite simple a bug in your maintenance script. Have you checked if there is a logical issue which causes your script to rebuild all indices instead of just those who matches the criteria. This maybe can cause it too.

Just beside all of this possibilities, it may be useful to check the log files for some changes to the options affinity mask, affinity I/O mask and their x64 partners. Another thing can be a change of the MAXDOP option of your instance. Please check the logs for them too. They will need to flush the plancache too.

Last but not least, you can still run a serverside trace (just setting it up using the profiler, start it, stop it and using sql command to start it again on the serverside). Beside that perfmon is your friend. It can watch and monitor your performance values for a time. Maybe you can see parallels in the pressure with certain actions on your server which may cause those flush.

Hopefully this will help you, even if the answer come a bit later.

Assuming you want randomish ordering of your output you can just ORDER BY NEWID(). That would jumble up the ordering of the results reasonably well. If you only need to randomize after a certain order is established then you can still use the NEWID() trick. Below are some quick examples:

-- Fully random ordering
SELECT
    *
FROM dbo.Foo F
    INNER JOIN dbo.Bar B ON B.Id = F.FkId
WHERE
    B.ImportantFilter > @ImpressiveVariable
ORDER BY
    NEWID() ;

-- Random ordering of results but the results still preserve the order of UserId
SELECT
    *
FROM dbo.Foo F
    INNER JOIN dbo.Bar B ON B.Id = F.FkId
WHERE
    B.ImportantFilter > @ImpressiveVariable
ORDER BY
    F.UserId
    , NEWID() ;

Another way, which will not produce all possible orderings but only a limited set but can be more efficient with large tables because it doesn't use NEWID(), only a single value for @aRandomNumber which should be drawn from the range of the column that is compared against (F.UserId here) :

WITH cte AS
  ( SELECT *, 0 AS ord
    FROM dbo.Foo F
        INNER JOIN dbo.Bar B ON B.Id = F.FkId
    WHERE ( <where conditions> )
      AND F.UserId >= @aRandomNumber
    ORDER BY F.UserId
  UNION ALL
    SELECT *, 1 AS ord
    FROM dbo.Foo F
        INNER JOIN dbo.Bar B ON B.Id = F.FkId
    WHERE ( <where conditions> )
      AND F.UserId < @aRandomNumber
    ORDER BY F.UserId
  )
SELECT <list of columns except the "ord">
FROM cte
ORDER BY ord, UserId ;