In 0.4 Earth's Shipyard could take population from Mars as well as Earth. However, Mars sent it's production, including from Asteroids that were sending production to Mars, to Earth and from Earth it got sent to the Shipyard. If Mars was a Colony it did NOT have it's own Shipyard, and making Mars into a CORE world is a BAD idea in 0.4 as well as in 0.45, Earth needs the additional production from Mars for Earth's Shipyard.
In 0.4 the Colony Worlds were often growing to 3 and 4 population, and were consuming a LOT of your total Food supply to feed them. And there was no way to avoid it since you would keep taking control (annexing) of the colonies that the AI Races kept colonizing in your Home Sector, via Influence. In fact, taking control of those many small Colonies is the fastest way to increase the various productions of your Core Worlds, if you can keep them supplied with enough Food.
In 0.4 I kept looking for a way for Earth's Shipyard to take population from Artemis, another small World that could be colonized in the Sol system, but this was NOT possible to do in 0.4.
In 0.45 the Colonies have stopped growing their populations large enough that you need or are able, to keep taking population from them and putting that population on Colony Transports and Constructors.
In 0.45 you can NO LONGER take Population for a newly launched Colony Ship or Constructor, for a nearby Core World's Shipyard, e.g. Earth, from any Colony World.
In 0.45 the Colonies no longer have the extra population that we were using for the Core Worlds', such as Earth, Shipyards.
This means in 0.45, that growing your population fast enough to keep supplying newly launched Colony Ships and Construction Ships is a bigger problem than it was in 0.4. You are going to need to work on ways to increase the population growth rate on your Major Core Worlds that are building a lot of Colony Transports, Constructors, and eventually (when Wars start) Troop Transports.
My personal experience in 0.45 is that early in the game, I rely more on getting the Drengen Prison Ship Survey result and turning them into my Colony Ships, with a Drengen population unit onboard it, rather than building a lot of my own Colony Ships. The biggest problem with those ships, is that they do NOT have Infinite Range, and once you start exploring/surveying other Sectors on a large scale, those new (Drengen) Colony Ships are often out of supply Range.
added: this leads to those ships having no choice in route to in-supply territory, and as just happened to me, the new Drengen Colony ship runs close enough to the Space Monsters that they kill it. It also means that they can not be used to colonize anything, until they are in supply.
It takes awhile before you can research and build things to increase your population growth rate on your Core Worlds, since you have many other things that need to be researched also.
A longer term solution, is something that I realized can be built in GC IV: Once you have BOTH the ability to build Troop Modules (from Planetary Invasion Tech) and can build Medium Hull Ships, you can design and build a Medium Hull ship with BOTH a Colony Module and a Troop Module. This takes 6 Slots, so you can NOT do it with a Cargo Hull ship design which only has 5 Slots.
You are NOT allowed to have 2 Colony Modules, or Troop Modules, or Construction Modules on the same ship design in GC IV, even if it has the needed 6 slots.
I call the design a Passenger Ship. It HAS to carry at least ONE population unit when you launch it from a Shipyard or a World or a Star Base. However, it can carry TWO population units as soldiers. This gives you the ability to move a net one population point per round trip in either direction, i.e. you can launch from Earth's Shipyard with one population point from Earth, and move to another Core world, with at least 2 population, and land there. Then you can launch the Passenger ship carrying Two population (as Soldiers) and send the Passenger Ship back to Earth, increasing Earth's population by a net one point, assuming that the Passenger ship is going to be making future trips.
Building a Passenger Ship is only worth doing if you are going to use it multiple times to move population, otherwise just use the old ships.
You could do something similar to this in GC III, where a Colony Module HAD to carry ONE population point when it was launched, but it COULD carry THREE population points, for a net transfer of 2 population points. I did this a lot in GC III. You could even design and build a Cargo Hull Ship with TWO Colony Modules in GC III and carry 1 to 6 population points on it.
This Passenger Ship design is awkward to use, but it is the only way to shift population from World to World in GC IV, besides a Colony/Constructor/Troop Ship that moves some where, and then STAYS there.
I have just thought of another two possibilities, After you have Medium Hull Tech, you "Might" be able to build a Medium Hull Ship with one Colony Module and one Constructor Module which could carry two population units. This design would not need the Tech to build Troop Modules.
You "Might" even be able to design and build a Large Hull Ship (that has 10 slots) that carries a Colony Module, a Construction Module, and a Troop Module, that could carry 1 to 3 population points.
I have not (yet) tried the last two ideas, but they will probably work.