Saving humans is honorable task and not so often is asked here at WB, so it's worth to think about.
Keltari provided pretty solid answer why not, in current state of everything.
Although our current state of everything is far from perfect, and beyond our current demands. Besides floating islands happens cataclysms, affecting even more peoples. Tsunami Mortality Estimates and Vulnerability Mapping
in Aceh, Indonesia
Approximately 1 year after the tsunami, Indonesian government estimates totaled 129775 deaths, 38786 missing, and 504518 tsunami-displaced persons in Aceh Province.
Some countries have special forces to act in such situations. So it's safe to assume to have such force on Earth level, as UN squad with top notch technologies, which actions might be justified world wide, after actions have took place.
For such limited time, ability to make fast decisions is crucial.
What we are talking about, island
- land area 151 $\small km^2$
- population 470000
Besides the fact our current big(more then 10k peoples) cities are not suited for rapid evacuation, this place is even worse, only 4 bridges connects it to mainland. Without panicking, with strict order 4 peoples per car, 6m per car (bumper bumper traffic) with average speed 40 km/h, 4 directions - evacuation capabilities are 30 peoples/sec, 4.5 hours evacuation time - not bad actually, and this is with one line traffic, and we have at least 2, and Verrazano-Narrow Bridge have at leas 6 lanes as I may see.
Way much better then I initially have assumed.
So 70000 tesla autopilot-ed cars may evacuate 470k peoples in 40 minutes - 20 min in, 20 min out, at 120km/h speed, 7 peoples per car. At this scenario peoples themselfs, their will to cooperate will be a problem, immobile peoples problem also not solved, bridges have to be intact, but for 99% existing technologies isn't so bad. Tesla have to think about emergency evacuation plan firmware for their cars.
- I will focus on evacuation, from non sapient form of danger, where people are not aware about that danger.
- I will not pay attention for those who are in nuclear bunkers at that time, their evacuation can be done same way, just more details needed to describe how.
- I will try to keep it as description of use case, not how it works, more details on technology level involved may be read here, paragraph "Note about Venus scrap, snake elephant". For those who not interested in reading, may take look at that video, to get rasp how it works: STRANGE but GENIUS Caterpillar Speed Trick - Smarter Every Day 93 (it worth looking by itself)
- for those who worrying about algorithms and programming, I'm happy to introduce this creature Slime mold form a map of the Tokyo-area railway system, this is also interesting specially begin part Are You Smarter Than A Slime Mold? and this one also interesting This Pulsating Slime Mold Comes in Peace
So algorithms may be simply enough without involving some special programming for that particular case.
- for color changing stuff (displays and so on) This is a butterfly (Scanning Electron Microscope) - Part 2
Situation is same as for slime, peoples are point of interests, who have to be transported in minimum time and effort to save location.
As human behavior is main problem in evacuation we have to force them for proper one, but to keep hassle at minimum in after evacuation time we will evacuate houses, trees, grass, pets, rats, worms, everything.
Let say 10m deep layer of that island. Ok it can be done with tool. But let focus on only human evacuation, it's more interesting.
Let assume UN emergency base are 5000km or less away from location, this is approximately 10 bases per Earth, or less in case continents only. To implement our plan We have to drop, let say per each square km, 2000t (2m tick layer which may cover that square km) of our technological slime, or 300k tonnes of it per island, it's like a big tanker of slime.
4 bases may take actions, to launch something like 100k tonnes per each base - launching is done Launch loop style. (not literary by Launch loop, just principle)
This is kinda tricky part, because we have to deploy tool to place in less then 10 minutes, but having tool and not having plan about rapid deploy makes no sense, but it can be done, just 30M speed. Also not having energy for that situation also makes no sense, it have to be stored long before actions, in form of kinetic energy or something else.
Altitude of transporting is something like 20-30km - to be able to use atmosphere to prevent tool to orbit or earth escape - and have control about that part of problem.
After arrival on place part have to be released at altitudes 20-30 km over the place.
They may form flying mat - and descent an needed speed profile and cover entry island. Searching each hole for human presence and puppies. and cats. and any living creature from carbon and water food included.
Detection of such stuff can be done automatically, without human interactions, it's kinda how geologist's searching for valuable materials with help of neutron sources, with MRI
Each base have to have personnel which have to be involved in to actions, and take special attentions to places like hospitals, data centers, library's, research facilities, nuclear reactors, chemical facilities, flying jets, airplanes, helicopters.
It's enough to assign 1 from personnel per 10-20 evacuees, 50-100 from personnel per each place that demands higher attention.
Each found carbon-water based object - have to be encapsulated in shell, supplied with oxygen removed CO2 if it produces it.
Each object with mass above 2kg have to receive message about reasons and short explanation why evacuation is happening, with promise to explain future information later in near future.
All objects mass 0.5 kg who produces CO2 and higher have to be checked possibly being humans. (actually it's not hard task to distinguish human body from all others objects)
Humans object have to be assigned priority over all other objects and acceleration limit to 1-2g, special orientation in space, live support (like space suit), monitoring of vital function, engaging combatant live support algorithms in case of demand(artificial air breading, heart stimulation, blood cleaning, etc all kinda paramedic stuff which I'm not familiar with but they keep lives with very limited resources - tool can do better, way much better).
So real tricky part will be something like ongoing delicate operations, on brains and such, one of possible solutions evacuate building - replace fundamental with tool, make connections electricity etc, or evacuate this room, may be more tricky. But evacuating entry medical facility makes sense. It have to build tool way in first place, but tool non distribution treaty and that political nonsense u know.
10 bases with 5000 personnel each can totally take control over situation and pay pretty close enough attention for each human found.
Tool can form VR goggles, transfer data and electricity, air water - so informing and controlling humans will be possible task. Also it will help to prevent some claustrophobic effects, injuries connected with panic reaction.
That also allows to make avatars for bases personnel, so they may act usual human way in situation where standard algorithms not working (there are also another advantages in that - algorithms can be fast programmed by human actions, like to days CGI effects but better and not about CGI effects.)
After collecting - tool, technological slime spit out capsules with evacuees and capsules fly to UN base in Sahara desert, probably at ballistic missile speeds, or aircraft style - depends on what is better in that situation. Or form sort of floating mat like ants do, with train lines with how much lanes we need.
ask in comments if something not clear, do not know what to add more, big task and there are lots of details, but general picture have to be more or less clear. I have notice it's not grey goo it's less advanced technology, but many stuff gree goo may do, tool may do also - almost everything at 0.1mkM scale with some limitations, but at least in this case abilities have some similarities.