As far as I know, photons' wavelengths can be considered increasing as space expands, making them lose energy and momentum. Does the same apply to physical objects? I understand a photon's speed is the same regardless of the system of reference, and the same does not hold for physical objects, but is it possible to draw any kind of analogy, for instance, would the momentum of an object launched from Earth (neglect non-cosmological effects) be steadily decreasing when looked at from Earth?
GR doesn't have global frames of reference, so we can't say whether a projectile launched from galaxy A slows down relative to galaxy A due to cosmological expansion when it's at a cosmological distance. However, suppose that galaxy A and galaxy B are both at rest relative to the Hubble flow. We can ask whether the velocity of the projectile relative to B, when it gets to B, is lower than its velocity had been relative to A, when launched from A.
There are a couple of easy ways to see that the answer is yes.
One is to consider the fact that ultrarelativistic massive particles have to have same behavior as massless particles. For example, people didn't even used to know that neutrinos had mass. So an ultrarelativistic neutrino, just like a photon, has to lose momentum and energy by the time it gets to B. If this holds for ultrarelativistic particles that have mass, then we expect it to hold as well for lower-energy particles that have mass, because we expect the behavior to vary smoothly with energy.
Another way to see this is that we know the universe cooled down as it expanded. This means that massive particles must have lost energy. We can't blame this on interactions, because actually the matter in standard cosmological models is an ideal gas. So the result must be the same, on the average, for a particle that just travels freely. If there was no such tendency for motion to settle down to the Hubble flow, then we wouldn't have a Hubble flow now.
It's not true, however, that redshift factors are the same for ultrarelativistic particles as for nonrelativistic ones. The effect is bigger for ultrarelativistic particles, which is why the universe is no longer dominated by radiation, even though it was at one time.
For simplicity I'll refer to the launched object as a probe and other cosmological objects as planets (one of which is Earth).
There's no good answer to what happens to the probe's speed relative to Earth, since there's no good way to define a notion of relative speed of very distant objects in cosmology.
The speed of the probe relative to the nearest planet will decrease over time. This happens simply because the planets are moving away from each other. If you imagine the planets and probe to have negligible mass so that there is no gravity (and also no cosmological constant), everything has a constant speed, and the probe will eventually pass every planet that has a lower speed, and will never pass any planet that has a higher speed, so ultimately it will end up permanently in between planets with slightly higher and slightly lower speeds, with a small speed relative to them. To put it another way, objects that initially have a high peculiar speed relative to the Hubble flow end up moving with the Hubble flow, if you wait long enough.
Cosmological redshift happens for the same reason. You can imagine that the light is absorbed/detected by each planet it reaches and then reemitted at the same frequency. The reemitted light will be detected by the next planet with a redshift or a blueshift depending on the relative motion of those two planets. On average, the planets are moving apart, so the longer the chain of planets, the larger the accumulated redshift.
The energy density of the universe in the present era is very low, so this gravityless model is pretty accurate out to distances of hundreds of millions of light years. At larger scales, you can no longer ignore spacetime curvature, but spacetime curvature doesn't fundamentally change what happens, it only deforms it a bit. It's a misconception that the loss of momentum is due to some peculiar general-relativistic property of spacetime, like curvature or intrinsic expansion. It's simply due to the fact that the planets (and stars and galaxies) are moving away from each other.