It is not possible for a sequence to "converge" to one thing and then to another. The higher-order terms in an asymptotic expansion will go to zero. What they tell you is how close to zero they are for any given value of $n$.
For the Central Limit Theorem (as an example) the appropriate expansion is that of the logarithm of the characteristic function: the cumulant generating function (cgf). Standardization of the distributions fixes the zeroth, first, and second terms of the cgf. The remaining terms, whose coefficients are the cumulants, depend on $n$ in an orderly way. The standardization that occurs in the CLT (dividing the sum of $n$ random variables by something proportional to $n^{1/2}$--without which convergence will not occur) causes the $m^\text{th}$ cumulant--which after all depends on $m^\text{th}$ moments--to be divided by $(n^{1/2})^m = n^{m/2}$, but at the same time because we are summing $n$ terms, the net result is that the $m^\text{th}$ order term is proportional to $n/n^{m/2} = n^{-(m-2)/2}$. Thus the third cumulant of the standardized sum is proportional to $1/n^{1/2}$, the fourth cumulant is proportional to $1/n$, and so on. These are the higher-order terms. (For details, see this paper of Yuval Filmus for example.)
In general, a high negative power of $n$ is much smaller than a low negative power.
We can always be assured of this by taking a sufficiently large value of $n$. Thus, for really large $n$ we can neglect all negative powers of $n$: they converge to zero. Along the way to convergence, departures from the ultimate limit are measured with increasing accuracy by the additional terms: the $1/n^{1/2}$ term is an initial "correction," or departure from the limiting value; the next $1/n$ term is a smaller, more quickly-vanishing correction added to that, and so on. In brief, the additional terms give you a picture of how quickly the sequence converges to its limit.
These additional terms can help us make corrections for finite (usually small) values of $n$. They show up all the time in this regard, such as Chen's modification of the t-test, which exploits the third-order ($1/n^{1/2}$) term.