We present a study of the mean transverse momentum $⟨p_{\mathrm{T}}⟩$ of identified strange hadrons (${\mathrm{K}}_{\mathrm{S}}^0,\Lambda ,\overline{\Lambda },{\Xi }^{-},{\overline{\Xi }}^{+},\varphi ,{\Omega }^{-},{\overline{\Omega }}^{+}$) produced in Au+Au collisions at RHIC-BES energies (the nucleon–nucleon center-of-mass energy $\sqrt{s_{\mathrm{N}\mathrm{N}}}=7.7 \mathrm{G}\mathrm{e}\mathrm{V},11.5 \mathrm{G}\mathrm{e}\mathrm{V},19.6 \mathrm{G}\mathrm{e}\mathrm{V},27 \mathrm{G}\mathrm{e}\mathrm{V}$ and $39 \mathrm{G}\mathrm{e}\mathrm{V}$). The mean transverse momentum is obtained from transverse momentum spectra of the strange hadrons as measured by the STAR experiment and its dependence on the number of participants $N_{\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}}$ is studied. For RHIC-BES energies, experimental data indicate a centrality dependence of $⟨p_{\mathrm{T}}⟩$, with an increase towards central collisions. This dependency is described using a power-law function to fit the data. The power-law exponent $\alpha$ is used to characterize the degree of flattening of $⟨p_{\mathrm{T}}⟩$ with respect to $N_{\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}}$ and its dependency on the collision energy and particle mass is studied. Special emphasis is placed on $\varphi$-meson that has a smaller interaction cross-section, thus reflecting the properties of the early stages of the system’s evolution. The $⟨p_{\mathrm{T}}⟩$ of $\varphi$-mesons produced in Au+Au collisions at RHIC-BES energies are compared with the results obtained in Au+Au collisions at higher RHIC energies and in Pb+Pb collisions at SPS and LHC energies. A distinct energy dependence of $\varphi ⟨p_{\mathrm{T}}⟩$ values is identified. Furthermore, data indicate, when comparing peripheral and central heavy-ion collisions, that $\varphi$-meson $⟨p_{\mathrm{T}}⟩$ increases with system size, following two distinct trends. The results are compared with the predictions of the default and string-melting versions of the AMPT generator. We observe that the string-melting AMPT version describes the strange meson $⟨p_{\mathrm{T}}⟩$, but underpredicts the strange baryon $⟨p_{\mathrm{T}}⟩$ centrality dependence. The default AMPT overpredicts the ${\mathrm{K}}_{\mathrm{S}}^0$ and $\varphi$ meson $⟨p_{\mathrm{T}}⟩$ centrality dependence, while the strange baryon data are in general better described by this version of the model. The exponent $\alpha$ obtained from AMPT-simulated results does not describe the measurements satisfactorily.