It seems that your first bullet point was precisely what I was after. Thank you very much!

@MartinHairer Thanks for the reply! My broad question would be: is there a way to normalize $(2)$ so that, as a function of $t$, it converges to a non-trivial limit? My motivation comes from the case $\bar t=0$, where $(2)$ converges to a time-changed Brownian motion (rather than a time-changed Brownian bridge).

@Amir For every value of $i$. I edited the original post accordingly.

@michael You are correct, and indeed I am essentially looking for a way to re-normalize (2) so that it converges to "something".

My guess would be that it should be provable for the space of cadlag functions on $[0,\infty)$ with some Skorohod topology (possibly $J$).

@DavideGiraudo Thank you for your suggestion. It was helpful. It does seem, however, that the results there are more suited for compactly supported processes and not processes on $[0,+\infty)$ (he uses finite partitions of the parameter space $T$).

What do you mean by stationary arrival process? A Poisson process has stationary increments, but it's not itself stationary (meaning: the distribution of $N(t)$ is different from the distribution of $N(t+h)$).

Hi @James ! Thank you for your answer. The answer itself is very clear, but I am not familiar with the technique, so I have a a question. It looks like you are proving that for every $t$ there exists a coupling such that the queues with $\lambda_n$ input and a queue with $\lambda$ input agree from $t$ on with high probability.. How does stationarity enter the picture here?