  We unravel the periodically driven dynamics of two repulsively interacting
bosonic impurities within a bosonic bath upon considering either the impact of
a finite pulse or a continuous shaking of the impurities harmonic trap.
Following a pulse driving of initially miscible components we reveal a variety
of dynamical response regimes depending on the driving frequency. At resonant
drivings the impurities decouple from their host while if exposed to a high
frequency driving they remain trapped in the bosonic gas. For continuous
shaking we showcase that in the resonantly driven regime the impurities
oscillate back and forth within and outside the bosonic medium. In all cases,
the bosonic bath is perturbed performing a collective dipole motion. Referring
to an immiscible initial state we unveil that for moderate driving frequencies
the impurities feature a dispersive behavior whilst for a high frequency
driving they oscillate around the edges of the Thomas-Fermi background. Energy
transfer processes from the impurities to their environment are encountered,
especially for large driving frequencies. Additionally, coherence losses
develop in the course of the evolution with the impurities predominantly moving
as a pair.
