I haven't found any primary sources yet. However it is a bit circumstantial but ...
The Complete Baronetage, vol. 2, p. 256 states that Walter Esmonde the 6th Baronet, married after Jan 1722, Joanna ... 2nd dau of Theobald Butler, 7th Baron Cahir with his 1st wife Mary Everard.
The Complete Peerage, 2nd edn, vol. 2, p. 467 doesn't have any dates for Theobald Butler's first marriage but does state that he married his second wife in Feb 1693, which means Joanna, the wife of Walter Esmonde, 6th Bt had to be born about 1692 or earlier.
Back to The Complete Baronetage, it states that Laurence Esmonde the 3rd Baronet, married in 1703 and died about 1717. Following thePeerage, If Walter, 6th Bt was his third son, then he would be younger than his wife by perhaps about 15 years or more and would probably be married as a minor (depending on how much after Jan 1722 the marriage took place).
If though following Cokayne and Burke's Peerage, Walter. 6th Bt was the son of Laurence Esmonde, 2nd Bt (see p. 255) who died in 1688, then he would be about the same age as his wife and married when he is an adult.
As I said it's circumstantial but I think it is more likely that Cokayne and Burke's Peerage are correct and John and Walter, 5th and 6th Bts are the brothers of 3rd Baronet not his sons.