I believe most law firms in CT do this now.  In a news article this week, a California attorney called the way CT handles real estate closings "backward and out of sync with the rest of the country." Lucky thing for CT's abstractors who work for in-state title companies and lawyers.

A VERY GENERAL example:  Because it is more cost effective and faster, abstract ordered through AAA title insurance company (attorneys here WRITE and sign the title insurance policies as agents for the company).  Abstract arrives with bill. Law firm to pay title company; however, we have name, address and phone #s for abstractor contracted through title company if there is a question on search.  Separate line item on HUD for abstract listing title company as payee and separate line items listing owners and lenders policy premiums, which includes disclosure of % of premium law firm that wrote the policy receives.  Depending on how soon from receipt of abstract the deal closes, abstractor cost may be paid prior to closing or, written through clients trust account, at closing.  Either way the law firm collects the abstractor cost from its client.

As someone who wants to see abstractors paid, I think "backward and out of sync" may be a good thing.