There was a great discussion on this topic at the recent Annual Meeting of the American Dairy Science Association (June 25-29, Pittsburgh, PA).  Several experts on the subject presented impressive data (historical, current and likely trends in the future) indicating that there is further potential for productivity improvements in Holstein dairy cattle in North America.  You could read one of the presentations by Prof. Jack Britt in the following link.

https://www.researchgate.net/project/Dairy-industry-in-the-USA-in-2066--50-years-from-now

As for enteric methane production, according to published research, intensity of CH4 emissions (g CH4/kg of energy corrected milk produced) continue to decrease with increasing milk production as a result of more feed energy partitioning to milk production while energy required for maintenance remaining more or less constant.  However, I too believe that there is a limit for this trend, both in terms of animal welfare and the environmental impact related to other factors (e.g. N, P excretion…).  

This is a good question and many wonder if we are reaching the biological limit. The answer seems to be that we still have a lot of room for increased yield. The top yielding cow in the USA during the past year produced 77,000 lbs (35,000 kg) of milk with 2X per day milking in one year. Her yield was 12 Standard Deviations above the average cow in the USA. We now have herds that are producing an average of 20,454 kg per cow per year. Top geneticists from Australia, Ireland and the USA reported at the recent American Dairy Science Association  meeting that there is no evidence of a plateau or limit in potential yield with the current population of Holstein cows. Remember that heritability of yield is about 0,25, so 75% of the variation in yield is associated with how the herd is fed and managed. I am aware of one herd in Wisconsin where the 3rd-5th lactation cows are producing an average of 73 kg per day (160 lbs per day). Thanks for your interest in this topic.

In regard to the question about methane production, the answer is fairly clear. Higher producing cows produce more total methane, but they produce less methane per kg of milk. As production goes up, the methane output per liter of milk goes down and this is consistent among breeds in countries worldwide. So to reduce methane output from dairy cows, we need more higher yielding cows and fewer lower yielding cows. Among all cows in the world (approximately 274 million), average yield is approximately 2,240 liters per year. Among the top 50 countries ranked by yield, there are 58.8 million cows averaging 6,932 liters per cow per year. Some countries with the most milk cows produce under 400 liters per cow per year.

Thank you for your responses. There has been ongoing debate in New Zealand about how long we can continue to improve our per cow production in pastoral agriculture. We have had a steady 2 percent per annum improvement in methane emissions per kg of milk over the last 25 years. Many in NZ argue that we might be reaching a plateau. I have argued that based on the potential of EU and US animals we have a long way to go before we reach any plateau.