How do space whales move? With great difficulty is the simple answer. The medium of space is too low density for an astrocetacean to swim in any way that resembles the motion of a whale in the ocean. Relying on conventional physics this only leads rocket propulsion.
The higher the velocity a space whale wants to attain it must exhaust large amounts of reaction mass or its exhaust velocity must be sufficiently high. if a space whale wants to reach a given velocity equivalent to its exhaust velocity then its mass ratio, which the ratio of reaction mass expelled over its final mass, will be approximately a factor of two. Note: this is an approximation value used for illustrative purposes where precision is not a requirement.
Chemical rockets have exhaust velocities of about four (4) km/s; a fusion rocket will be roughly 0.1 c (or 30,000 km/s); photon rockets is 1 c.
For a space whale to move it needs to have a propulsion system incorporated into its body. Chemical rockets are moderately feasible for a living creature (but with lost of caveats). The main drawback is it will take a space whale extremely long time periods for it to travel anywhere, even in a planetary system, let alone interstellar distances where the mass ratio would be massively prohibitive.
Chemically propelled space whale might start off with the mass of a blue whale but it would be reduced to the size of a sardine after boosting to its cruise velocity for modest interstellar trip. Many interplanetary excursions wouldn't be much better.
Fusion propulsion means space whales would need to incorporate fusion reactor technology into their bodies. A space whale now resembles what it will need to be, namely, a cyborg spacecraft. Fusion propulsion will involve low rates of acceleration. Roughly one centimetre per second squared. Interplanetary travel is feasible, while interstellar travel will be at its limits and probably infeasible.
Photonic propulsion requires incorporating serious technology into a cyborg space whale (it is unlikely there will be any other kind) and extremely dangerous too. They effectively need antimatter power systems to make them work well enough to be useful. So unless someone is prepared to feed space whales antimatter this is improbable.
Space whales are part of a highly advanced galactic civilization. Undoubtedly they would to be created from a combination of synthetic biology and cyborgization. This civilization has fast FTL travel technology. Possibly, the creators of the space whales will have equipped their space whales with FTL drive-systems. Once again whatever power systems are required will have to be part of their bodies.
This will enable the space whales to travel rapidly from one location hospitable to their survival to another. Otherwise their travel times will too long for their survival. Although space whales may need to go into a state of cryptobiosis while in transit.
The next level of locomotion for space whales requires consideration of exotic physics. The best example of which is the application of negative mass.
Although no particles are known to have negative mass, physicists
(primarily Hermann Bondi in 1957,[5] William B. Bonnor in 1989,[11]
then Robert L. Forward[12]) have been able to describe some of the
anticipated properties such particles may have. Assuming that all
three concepts of mass are equivalent the gravitational interactions
between masses of arbitrary sign can be explored, based on the
Einstein field equations:
Positive mass attracts both other positive masses and negative masses.
Negative mass repels both other negative masses and positive masses.
For two positive masses, nothing changes and there is a gravitational
pull on each other causing an attraction. Two negative masses would
repel because of their negative inertial masses. For different signs
however, there is a push that repels the positive mass from the
negative mass, and a pull that attracts the negative mass towards the
positive one at the same time.
Hence Bondi pointed out that two objects of equal and opposite mass
would produce a constant acceleration of the system towards the
positive-mass object,[5] an effect called "runaway motion" by Bonnor
who disregarded its physical existence, stating: “ I regard the
runaway (or self-accelerating) motion […] so preposterous that I
prefer to rule it out by supposing that inertial mass is all positive
or all negative. ” — William B. Bonnor, in Negative mass in general
relativity.[11]
Such a couple of objects would accelerate without limit (except
relativistic one); however, the total mass, momentum and energy of the
system would remain 0.
This behavior is completely inconsistent with a common-sense approach
and the expected behaviour of 'normal' matter; but is completely
mathematically consistent and introduces no violation of conservation
of momentum or energy. If the masses are equal in magnitude but
opposite in sign, then the momentum of the system remains zero if they
both travel together and accelerate together, no matter what their
speed:
The main problem with negative mass "runaway motion" drive systems is how to generate the negative mass necessary to propel a space whale. This may be cavalier, but a galactic civilization of the kind postulated by this question should have solved that problem long ago in its history. In fact, they might find it to be quaintly old-fashioned.
In summary, space whale will be cyborg spacecraft. They could be propelled by fusion rocket propulsion systems (a feasible future technology in terms of current science and engineering) or the more exotic negative mass drive system (conceptually plausible, but relies on the existence of the hitherto undiscovered negative mass).