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Say you've got a mothership that is on its way to a star. Could you launch a smaller ship from it while it's in either the coasting or braking phase of the journey? A scout ship, for instance. Or would the scout ship lose momentum too quickly and turn everyone inside into borscht?
I'm assuming a maximum speed of about .5c and a three phase journey: acceleration, coasting, and deceleration approaching target system.
I registered here to help the OP because these long answers don't seem to answer a major subtlety of the question:
As said you can leave in the coasting phase fine - inertia is the same.
However, in the braking phase - no, you could not assume it is possible to leave the mothership in a scout ship if the mothership was accelerating or decelerating at the time, it would immediately have a different relative velocity, because the scout ship's inertia would only be the same as that of the mothership at the moment of departure, and immediately the mothership acceleration or deceleration would become apparent, as soon as launch occurred. Whatever the rate acceleration or deceleration was at the time of it's leaving the mothership would appear to accelerate at that speed away from the scout ship.
However if the rate of acceleration or deceleration wasn't large it may be possible to leave and return literally "catching up" the mothership i.e. the only way for the scout ship to counter this is to apply the same (or greater) acceleration or deceleration to the scout ship which would take fuel.
Additionally if the mothership was sufficiently massive it could drag the scout ship in it's gravitational "wake" to some degree mitigating the relative velocity issues.
In the vacuum of the interstellar space, there is nothing to lose momentum to (unless you hit something, which is unlikely), so no, a scout ship wouldn't lose momentum too quickly and turn everyone inside into borscht. It would cruise on inertia just like its mothership, and move alongside it, unless some external force acts on it differently than the mothership (e.g. entry into an atmosphere, with it providing drag on the scout ship), or the scout ship exchanges momentum with some other matter (e.g. reaction mass of own propulsion system, gravity assist, magnetic or solar sail, and so on), or the mothership itself (some sort of repulsions, like magnetism, photon pressure,...).
This is actually a big problem for proposed methods of interstellar travel, with perhaps most feasible one that is also capable of traversing such immense distances in reasonable time being beam-powered propulsion, where the source of momentum stays at the departure point and the ship is propelled on the power of deflected photons ($\mathrm{F} = 2\times \mathrm{P} / c$, in Newtons, for 100% sail reflectance, where $\mathrm{P}$ is total reflected power from the source in Watts, and $c$ the speed of light. Divide that with craft's mass in kilograms and you have acceleration that it would get from it. E.g. a 100 kg sail reflecting 50 GW at 100% reflectance, would accelerate at 3.36 m/s², so about 1/3 the gravitational acceleration on the surface of the Earth).
That comes with its own problems, of course, and I won't go into too much detail about it here, but most such proposed designs also call for a decelerating sail detaching from the mothership and acting as a reflector to reverse direction of the beam and decelerate the mothership in the opposite vector relative to its motion (sometimes called a brachistochrone turn, since it would be a constant acceleration system, but it is a bit of a mouthful and, in earnest, I've still not mastered pronunciation of this term). Of course, the deceleration sail is still being propelled in the direction of the beam while it reflects it, so the distance between the mothership and the deceleration stage would constantly increase. And this is where exchanged momentum goes to decelerate the mothership in a system like that. It's just an example though, but I've picked it because it's directly applicable to your question and perhaps not as intuitive how momentum is exchanged in it, while also demonstrating the need for it.
Point is, that all Newton's laws of motion of course apply also in the interstellar space:
First law: When viewed in an inertial reference frame, an object either remains at rest or continues to move at a constant velocity, unless acted upon by an external force.
Second law: The vector sum of the external forces $\mathrm{F}$ on an object is equal to the mass $m$ of that object multiplied by the acceleration vector $\mathrm{a}$ of the object: $\mathrm{F} = m\mathrm{a}$.
Third law: When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body.
Consequence of that is that a smaller mass body like a scout ship detached from the mothership will require less force to decelerate, or conversely it would decelerate faster at same force applied in the vector opposite to its direction of movement than a more massive body, but unless it's acted upon by some external force, this won't happen and will just cruise alongside its mothership until something changes for it with respect to the mothership.
They would likely already have different mass, so you could then apply same source of decelerating force to each differently (e.g. with surface per mass for passive sails, burn rate per mass for own propulsion,...) and they will start moving away from each other. But the force of this decelerating force acting on its inhabitants would likely be small relative to its net momentum, so acceleration felt by its inhabitants would also be small. Again, unless you hit a solid body at largely different relative velocity. Then, they would turn into borscht and the energy released upon impact equals $E_\text{k} =\tfrac{1}{2} mv^2$, so as you can see, relative velocity matters a whole lot more than the mass of the object, tho it's still relevant, too. But that's a different matter (no pun intended).
First off, space is mostly empty, and simply dropping a probe will only lead to the mothership and the probe flying in formation (during coasting phase). During braking phase the probe will fly forward happily without mechanical complications (just remember to launch it from the side of the ship and maintain lateral separation).
Second, there are possible physical complications and they do actually depend on the speed of the mothership. A mothership going at an appreciably large speed relative to $c$, the speed of light, hits interstellar matter which gives rise to nasty Bremsstrahlung radiation. Protecting manned scout ships is costly in terms of mass. Of course, you don't want to launch a scout ship if it cannot decelerate to achieve orbit within the target solar system or around one of the planets. It may have to use either aerobraking in a planetary atmosphere or solar braking (if it is light enough) around the target star, in addition to whatever propellant it has on its own.
And here's the rub: aerobraking needs exact knowledge of the atmosphere's density and chemical properties profile otherwise the probe will either skip out and fail to decelerate to orbital velocity, never to return, or burn if the probe dips too low. The ability to aerobrake is furthermore limited by thermal shielding, maximum allowed internal temperatures, structural strength, and heat absorption/rejection capacity. One may envision aerobraking from 13 km/s in the near future, and perhaps a bit more later.
Thus, if you are really going there in a huge mothership, it makes sense to decelerate below the star's escape velocity, or even get into orbit around one of the planets and only then launch probes - first unmanned to scout particular planets, then a shuttle (most likely multistaged) to get exploration crew on the ground for some photo sessions and flag planting stuff.
Other answers adequately explain the problems with a manned scout ship, but it actually makes loads of sense to launch a small unmanned probe before the deceleration phase. It will speed through the system at interstellar velocities, but can collect various kinds of data on the system and transmit it back to the mothership, allowing last minute course corrections and so on, or even just giving the opportunity to explore some of the non-target bodies in the system, in terms of delta-v it's going to be much cheaper to launch a (fly-by) probe while still in interstellar space - you already have lots of velocity for getting places quickly.
Because the chance of a high speed probe being obliterated by debris is non-negligible, you would probably actually launch a swarm of small probes - they don't need to be very big, just various instruments a transmitter and limited attitude control.
