If you are playing to win, you tell them you currently make whatever salary you are seeking. When they ask you what sort of salary you need, say something along the lines of "Obviously, I want to take a step forward, but I am pretty flexible, so if you can at least match my current pay, then I would be open to accepting".
While this tactic does fall into the grey area of morality, it is quite effective.
This is where you say "my total compensation is around $x". To me, "total compensation" includes pay, benefits, bonuses, the dollar value I put on how short my commute is, rough cost of free food in the office, how nice my chair is, etc. etc.
I also don't think they'll be able to do an equifax/transunion lookup without your permission. And it's perfectly reasonable to say no to that, as credit checks are bad for your credit.
"Well, I guess Eqifax is not that accurate" and "Oops. Thanks for letting me know I don't want to work here" would both be reasonable responses. Most companies are not doing this.
While this tactic does fall into the grey area of morality, it is quite effective.