Fallacy when someone defines own criteria for arguments being valid/true?

In discussions, there will sometimes be someone (usually the one who initially proposed the topic) who will try to direct the conversation, but also, and most importantly, define his own criteria for arguments presented to valid or true—and of which is then imposed on others, for their arguments to be accepted.

If the arguments presented don't fit those personal criteria, the person's response will usually be something along the lines of "it doesn't fit criteria A, B, and C, that I said earlier, therefore, your argument is invalid."

There is no chance of winning in such a situation (especially if you didn't spot the intellectual dishonest foundation of the discussion in the first place). Even if you do meet the dishonest criteria, the response will usually be the addition of a new criterion, again, making your argument "invalid."

For example:

A: Okay, so I want to talk about X . I say that X does not exist. However, I want to put aside the research  by Dr. Sark. Also, anyone who wants to participate needs to present arguments that follow X, Y, A, B, C, D, E, F, G, H, I and Z. Thank you. Now let's begin.

B (failed to see fallacious foundation of discussion): Okay, well, in 2004, it was discovered that ....

A: No, you didn't follow criterion A that I said earlier. Your argument is invalid and won't be considered. Try again. X still doesn't exist.

C : Well, there are many examples of this, in ....

A: You also didn't follow criterion A, as well as D and H. You argument is also invalid. Why don't people present valid arguments?! X still doesn't exist!

What is the name of this fallacy? How can it be countered?

Consider the person who is putting forward the requirements as the one making the argument. They are arguing:

P1. In order for X to be true, A,B, and C must also be true.
P2. A, B, and C are not true.
C. Therefore, X is false.

This is not a fallacy; it is actually a valid argument structure. However, it is not necessarily sound. It is up to the arguer to demonstrate the soundness of the argument. We should not accept P1 and P2 without supporting evidence.

A common example of this in the religious debates community is Jesus' resurrection. I have heard many theists argue (please don't hold me to the exact wording, as the mention of this example is for academic purposes only and not to make or defend the argument) that in order to debunk/show unreasonable the resurrection, one must explain several things including 1) the empty tomb, 2) the fact that witnesses saw Jesus alive after death, ... and a couple of more. Of course, if this is simply a legend and the stories in the Bible are fiction (fictionalized), we don't have to explain any of this.

The bottom line, hold the one setting the criteria for truth accountable for providing evidence for their (implied) argument and don't accept it until they do to your satisfaction.

Even if you do meet the dishonest criteria, the response will usually be the addition of a new criterion, again, making your argument 'invalid.'

This is a clear example of moving the goalposts .

I do not see this as a fallacy.  It is setting the terms of the discussion. That’s fine, unless the proponent is also claiming it is a discussion to ascertain the truth of X. 

As a matter of personal experience in this situation, the proponent usually fails to define X and leaves it as a loosey-goosey statement that it is so imprecise it cannot be the basis for a reasonable discussion.

If this is a discussion in which you desire to engage (I would prefer to get a cup of coffee), then do so by drilling down into how the proponent came to that conclusion. Do not offer any contrary facts or arguments. Simply ask questions to expose the error(s) the proponent made in arriving at conclusion X.

Nothing wrong with setting rules and criteria before the debate but if they try to change then in the middle that would be changing the goal post.

I understand valid criteria for a discussion, but rather, I mean such an individual will use arbitrary and personally-defined criteria for truth (or validity of arguments/what is presented).

For example: 
Person 1:  "Your topic is interesting... According to a recent comprehensive and peer-reviewed 10-year study conducted by MIT, ..." 
OP:  "Was this study funded by a large multinational corporation?" 
Person 1:  "Yes, it lasted 10 years, after all. It cost 10 million to conduct." 
OP:  "If your study was funded by a large multinational, then it has no value. Moving on." 
Person 2:  "According to X, ...." 
OP:  "Again, multinational company funding. Not valid. 
Okay everyone, no valid proof has been presented—my proposition is still 100% valid! "

This is obviously an exaggeration, but essentially: usually, the person who proposed the topic, by implementing his own personal criteria around the discussion, then uses these as arbiters of truth—or what is a valid or invalid arguments/proof/etc. In the end—solely because of the pre-defined criteria—the original proposition remains "valid."

It has similarities with moving the goal posts, but instead, more like "setting up the goal posts 1 inch apart, before the game starts (so the ball can't even pass through)."