Typically, the nation that has been in power reacts in a somewhat negative way to convergence. Typically, this will take the form of calls for economic "warfare" against the rising country. We can see this today in the case of the United States and China. As China catches up to the US, some in the US see it as a threat and want to enact protectionist measures against China. The same sorts of calls were heard when Japan was catching up to the US 20 or 30 years ago.
Countries also react domestically. They tend to come to believe that there is a crisis and that changes must be made. There are often (as today in the US) calls for improved education so as to regain the lead that is seen as their birthright.
In general, countries that are ahead do not like convergence and try to prevent it from happening both through foreign policy and through domestic policies.