Recently, the Belle II collaboration announced the first measurement of the branching ratio $B\left(B^{+}\to K^{+}\nu \overline{\nu }\right)$ $$ \mathcal{B}\left({B}^{+}\to {K}^{+}\nu \overline{\nu}\right) $$ , which is found to be about 2.7σ higher than the Standard Model (SM) prediction. We decipher the data with two new physics scenarios: the underlying quark-level $b\to \mathrm{s\nu }\overline{\nu }$ $$ b\to s\nu \overline{\nu} $$ transition is, besides the SM contribution, further affected by heavy new mediators that are much heavier than the electroweak scale, or amended by an additional decay channel with undetected light final states like dark matter or axion-like particles. These two scenarios can be most conveniently analyzed in the SM effective field theory (SMEFT) and the dark SMEFT (DSMEFT) framework, respectively. We consider the flavour structures of the resulting effective operators to be either generic or satisfy the minimal flavour violation (MFV) hypothesis, both for the quark and lepton sectors. In the first scenario, once the MFV assumption is made, only one SM-like low-energy effective operator induced by the SMEFT dimension-six operators can account for the Belle II excess, whose parameter space is, however, excluded by the Belle upper bound of the branching ratio $B\left(B^0\to K^{\ast 0}\nu \overline{\nu }\right)$ $$ \mathcal{B}\left({B}^0\to {K}^{\ast 0}\nu \overline{\nu}\right) $$ . In the second scenario, it is found that the Belle II excess can be accommodated by 22 of the DSMEFT operators involving one or two scalar, fermionic, or vector dark matters as well as axion-like particles. These operators also receive dominant constraints from the B 0 → K *0 + inv and B s → inv decays. Once the MFV hypothesis is assumed, the number of viable operators is reduced to 14, and the B + → π + + inv and K + → π + + inv decays start to put further constraints on them. Within the parameter space allowed by all the current experimental data, the q 2 distributions of the B → K (*) + inv decays are then studied for each viable operator. We find that the resulting prediction of the operator $Q_{\mathrm{q\chi }}=\left({\overline{q}}_p{\gamma }_{\mu }{q}_r\right)\left(\overline{\chi }{\gamma }^{\mu }\chi \right)$ $$ {\mathcal{Q}}_{q\chi}=\left({\overline{q}}_p{\gamma}_{\mu }{q}_r\right)\left(\overline{\chi}{\gamma}^{\mu}\chi \right) $$ with a fermionic dark matter mass m χ ≈ 700 MeV can closely match the Belle II event distribution in the bins 2 ≤ q 2 ≤ 7 GeV2. In addition, we, for the first time, calculate systematically the longitudinal polarization fraction F L of K * in the B → K * + inv decays within the DLEFT. By combining the decay spectra and F L , almost all the DSMEFT operators are found to be distinguishable from each other. Finally, the future prospects at Belle II, CEPC and FCC-ee are also discussed for some of these FCNC processes.