Once the Brexit transition period ends on 31/12/20, the EU will presumably insist on much more stringent control for travel between the EU and the UK, which will be a "3rd country", so all stations served by Eurostar trains will require appropriate infrastructure and manned security barriers. This is likely to have a major impact on travel via the Channel Tunnel, and will effectively quash any significant potential expansion of Eurostar destinations that might have occurred if Brexit hadn't happened. I expect that security and document control arrangements will need to be similar to those required for travel across the eastern frontier of the EU into Belarus and Russia, albeit presumably with some visa-waiver arrangements. The EU will presumably also insist on strict controls for travel across the Irish Sea, including between Great Britain and the 6 counties.
The border for people between the Schengen area and the UK is effectively the same as the border between the UK and any other country, and the Schengen area and any other country. Lots of non-EU citizens and non-British citizens take the Eurostar and the entitlements for these people to be in France are entirely separate from the entitlements for those people to be in the UK. Therefore, there are secure fences and barriers blocking off physical access to the platforms and tracks which Eurostar uses, and nothing would need to change with regards to this.
Flights from Kenya and flights from Poland land into the same terminals in the UK and passengers arriving from these destinations are processed in exactly the same way. There may be some background intelligence carried out if someone of interest is on a specific flight, which perhaps is more likely to happen on flights from outside the EU, but otherwise passengers all go through the same experience regardless of their origin. For flights landing into Schengen airports, the same is true but with one difference - EU airports trust UK security screening so passengers arriving from the UK into Schengen can mingle with departing passengers without going through further security screening. But EU airports also trust USA security screening so the same applies to USA arrivals. At the end of the transition period, there will be changes to the documents required for British citizens to travel to the EU, but there will be no changes to the documents required for Americans or South Africans to travel to from the UK to the EU, so saying that "document control arrangements" would need to be upgraded is tantamount to saying that the UK currently does not enforce immigration control properly.
There will be no changes to the travel arrangements for people when crossing the Irish Sea.
What will change is the customs arrangements. Currently Eurostar arrivals basically go through the Blue customs channel that is found in UK airports, as all arrivals are from the EU. When the blue channel ceases to exist, then I guess there will now be red and green channels at Eurostar stations. Both the UK and mainland Europe countries currently take an intelligence-led approach to customs checks. At UK airports, passengers arriving from Spain mingle with passengers arriving from Thailand in the baggage reclaim area, and are then trusted to walk through the correct channel. At Schengen airports, passengers arriving from the UK mingle with passengers from Argentina at baggage reclaim and are also trusted to voluntarily make the appropriate customs declaration if necessary. There are random spot checks (which may be focussed on common arrival times for planes from Asia, etc.). The same approach can be taken for Eurostar arrivals.
Based on what is currently known, there would be new customs checks when crossing the Irish Sea.
With regards to security, if someone goes through security screening and that screening is thorough, then they are not a security risk regardless of their documentation or the goods they are transporting.
When ETIAS is introduced, the EU says there will be automatic kiosks at land borders for people who didn't know, or were unable, to apply for ETIAS online. These could be built at the UK Eurostar stations.
I did wonder if Eurostar and the border control teams could set up temporary arrangements, for things like big sporting events or festivals. If they could, then Liege Guillemins would be a possibility
Is there really much advantage to having temporary mobile border control at Liege when Brussels is not that far away, all trains have to pass through, and most likely any extra staff would just be taken from those stationed at Brussels?