jats:titleAbstract</jats:title>jats:pTheories of dark energy that affect the speed of gravitational waves jats:inline-formulajats:alternativesjats:tex-math$$c_{\textrm{GW}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msub mml:mic</mml:mi> mml:mtextGW</mml:mtext> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula> on cosmological scales naturally lead to a frequency-dependent transition of that speed close to the LIGO/Virgo/KAGRA (LVK) band. While observations such as GW170817 assure us that jats:inline-formulajats:alternativesjats:tex-math$$c_{\textrm{GW}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msub mml:mic</mml:mi> mml:mtextGW</mml:mtext> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula> is extremely close to the speed of light in the LVK band, a frequency-dependent transition below the LVK band is a smoking-gun signal for large classes of dynamical dark energy theories. Here we discuss (1) how the remnants of such a transition can be constrained with observations in the LVK band, (2) what signatures are associated with such a transition in the LISA band, and (3) how joint observations in the LVK and LISA bands allow us to place tight constraints on this transition and the underlying theories. We find that deviations of jats:inline-formulajats:alternativesjats:tex-math$$c_{\textrm{GW}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msub mml:mic</mml:mi> mml:mtextGW</mml:mtext> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula> can be constrained down to a level of jats:inline-formulajats:alternativesjats:tex-math$$\sim 10^{-17}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:mo∼</mml:mo> mml:msup mml:mn10</mml:mn> mml:mrow mml:mo-</mml:mo> mml:mn17</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> in the LVK jats:italicand</jats:italic> LISA bands even for mild frequency-dependence, much stronger than existing bounds for frequency-independent jats:inline-formulajats:alternativesjats:tex-math$$c_{\textrm{GW}}\ne c$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:msub mml:mic</mml:mi> mml:mtextGW</mml:mtext> </mml:msub> mml:mo≠</mml:mo> mml:mic</mml:mi> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>. We use the strain data from GW170817 to bound the deviation of jats:inline-formulajats:alternativesjats:tex-math$$c_{\textrm{GW}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msub mml:mic</mml:mi> mml:mtextGW</mml:mtext> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula> to be less than jats:inline-formulajats:alternativesjats:tex-math$$10^{-17}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msup mml:mn10</mml:mn> mml:mrow mml:mo-</mml:mo> mml:mn17</mml:mn> </mml:mrow> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula> at 100 Hz and less than jats:inline-formulajats:alternativesjats:tex-math$$10^{-18}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msup mml:mn10</mml:mn> mml:mrow mml:mo-</mml:mo> mml:mn18</mml:mn> </mml:mrow> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula> at 500 Hz. We also identify a particularly interesting type of transition in between the LVK and LISA bands and show how multi-band observations can constrain this further. Finally, we discuss what these current and forecasted constraints imply for the underlying dark energy theories. </jats:p>