Yes, we have quite a bit of information at least in the first 4-5 generations, including your line. I am 8th generation counting Thomas b 1740 & Mary Smith Carson as 1.
I have been working with several distant cousins on T & M this the last year. I am very busy this week/end but I am attaching a couple of web sites to allow you to browse what you can right now. This first one is by Molly Hager who is my 6th cousin through your Elizabeth's brother Joseph. I will send you the link that goes to T & M and you can click on the childrens blue name and go down the generations. This link cost nothing you just have to register to use it and you will not get junk mail as a result.
It seems Adam Co was a popular place for the 2nd generation. According to our records Joseph was the first there and actually left some children behind there. Some of his syblings followed. Joseph went on from there to Rushville, Rush Co. IN for a time then on to Franklin TWP, Marion Co. Just SE of Indianapolis. I visited his grave site there about 6 weeks ago. Another sister Margaret Carson Cloud was in Clinton Co., Ohio close to Adams and when her husband died followed her 4 sons to Grant Co. IN, about 40 miles from where I am today. I visited her grave about 2 weeks ago. I have also found the grave site this week of Mary Carson Vandolah just west of Cinci in Dilion, Dearborn Co, IN., Oakdale Cemetery whish has an on line site. I have pics of all these graves I will send you later. Mary was in Oh east of Cinci for a time also. I was born in Elyria OH, and raised in Vermilion by Cedar Point.
I remember vitality my grandmother talking about Bratton relations around Washington Co. My whole line stayed there until my father left.
Here is a site for the DNA testing if your curious about that. There is a long family tree listing there. I have recently done it and my 5th Cousin once removed from Elizabeth's brother John Clarkson Carson has done it. Thomas is C-3 if you go here. You will have to register for this site also. I am the new admin there for the Carson Surname.