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M&A Due Diligence Differences Amongst Big 4

I have an offer to join another Big 4 M&A analytics group. It came unexpectedly but I am happy it did. 

I let my current job know that I got this offer i was going to think about and now they are saying they will see about creating a similar position for me at the firm. 

What should I do here??

Does it matter which firm I stay with? Is one Big 4 better than another when it comes to this?