I don't fully understand your question to give a direct answer, but I think that's mostly due to a number of misconceptions you hold about the universe. So I'll just try to clear those up instead.

First of all, the big bang didn't happen somewhere, it happened everywhere. There's no "center" of the universe that everything is moving away from. The universe did not start from a point, it started from a small scale factor and the scale factor rapidly increased.

Assuming the universe is infinite (we're not sure), you could think of it as expontentially more infinities being created all the time. Say you take a 1-dimensional line and put tick marks 1cm apart, to positive and negative infinity. Then you multiply every tick mark position by 2. Two times infinity is still infinity, but the nature of the infinity has changed: it's scaled up by a factor of two. You could then subdivide it again, adding markings in between, 1 cm apart. You can think of these new markings as representing new centimeters which have appeared on your 1D line. This is happening to the universe in three dimensions.

Now, instead of doing this by discrete intervals, just do it constantly, so the scale factor is always expanding. Any two points are always moving apart from each other at the rate of this expansion, let's say 1cm/cm/sec. Now, introduce an attractive force between objects (e.g. gravity) which obeys the inverse square law. If the objects are sufficiently close to one another, they'll move towards each other at a rate faster than 1cm/sec, which would overcome the expansion force and allow them to remain together. In the real universe, the fundamental forces have this effect, and this is why we don't observe everything flying apart on local scales.

Now let's say that the maximum speed limit is 10cm/sec, and light travels at this speed across our 1D line. If we have two objects 2cm apart, then the distance between them will grow by 2cm in one second. In that time, light from each will have travelled the same distance and met one another. Now, they're 4cm apart, and in the next second will travel 4cm apart, twice as fast. Light can still make this journey. At the next tick, they're 8cm apart, and light still makes it. 2/16th of a second later and they'll be 10cm apart and moving away from each other at the speed of light.

At this point, no future emissions will reach them. However, we haven't been emitting discrete light pulses - galaxies continuously emit staggering amounts of light. So, the space between them is full of light which is constantly making the journey towards the other end. If no other factors were at play, they'd still be receiving old light from each other forever. But there are other factors at play: the inverse square law, combined with the increasing redshift, will eventually reduce the light to effectively nothing.

These factors are sufficient to extend the effective observation duration of distant objects well after they've crossed the lightspeed event horizon. To recontextualize this into our real universe, we can look at extremely distant objects like GN-z11, which is 32 billion light years away. If light were to re-tread this distance, it would require twice the current age of the universe to complete the journey. However, this light only took 13.4 billion years to reach us, because it was emitted when the universe was much closer together, and carried towards us on the "flow" of expanding space time, while the object was carried away from us on that same "flow" - faster than the speed of light.

Oh, and it's also worth noting that the rate of expansion wasn't always consistent. Check out this wiki: https://en.wikipedia.org/wiki/Inflation_(cosmology)