This paper investigates the impact of one-loop renormalisation group (RG) running and mixing of Wilson coefficients in the Standard Model Effective Field Theory (SMEFT) on Higgs production at the LHC, focusing on the processes $ H_j $, $ t\bar{t}H $, and HH. The study considers a subset of operators closed under QCD corrections and explores how different renormalisation scale choices affect key observables such as the Higgs transverse momentum and the Higgs pair invariant mass. The effects of RG running on the constraints on Wilson coefficients from current and future LHC measurements are also examined. The SMEFT Lagrangian includes dimension-6 operators that parameterise deviations from the Standard Model (SM) predictions. These operators are subject to RG running and mixing, governed by the anomalous dimension matrix. The RG equations for these operators are solved at one-loop level, and the running of Wilson coefficients is implemented through a reweighting technique based on cross-section results from MadGraph5_aMC@NLO. The study shows that RG running and mixing significantly affect the differential distributions of Higgs production processes. For example, the transverse momentum spectrum of the Higgs in gluon fusion and $ t\bar{t}H $ production, as well as the invariant mass of the Higgs pair in double Higgs production via gluon fusion, are sensitive to the scale choice. The impact of RG running is most pronounced for the chromomagnetic dipole operator, leading to variations in the cross-section of up to $ \mathcal{O}(50\%) $. The analysis also demonstrates that the inclusion of RG running effects can significantly modify the constraints on Wilson coefficients. For instance, the fixed-scale scenario, where the system is evolved down to $ \mu = M_H $, provides the most stringent constraints. The dynamical scale scenario, where the scale is chosen based on the physical observables, and the no-running scenario, where the scale is fixed at $ \mu_0 = 1 $ TeV, yield different results. The inclusion of both linear and quadratic terms in the SMEFT analysis leads to more pronounced effects, with the allowed regions in the parameter space being significantly rotated. The results show that the combination of different Higgs production processes, such as $ H_j $, $ t\bar{t}H $, and HH, helps to substantially reduce the allowed regions in the SMEFT parameter space. The inclusion of future HL-LHC projections further improves the constraints on Wilson coefficients, particularly for the Higgs self-coupling. The study highlights the importance of considering RG running effects in the interpretation of SMEFT data, as they can significantly impact the extraction of constraints on Wilson coefficients.This paper investigates the impact of one-loop renormalisation group (RG) running and mixing of Wilson coefficients in the Standard Model Effective Field Theory (SMEFT) on Higgs production at the LHC, focusing on the processes $ H_j $, $ t\bar{t}H $, and HH. The study considers a subset of operators closed under QCD corrections and explores how different renormalisation scale choices affect key observables such as the Higgs transverse momentum and the Higgs pair invariant mass. The effects of RG running on the constraints on Wilson coefficients from current and future LHC measurements are also examined. The SMEFT Lagrangian includes dimension-6 operators that parameterise deviations from the Standard Model (SM) predictions. These operators are subject to RG running and mixing, governed by the anomalous dimension matrix. The RG equations for these operators are solved at one-loop level, and the running of Wilson coefficients is implemented through a reweighting technique based on cross-section results from MadGraph5_aMC@NLO. The study shows that RG running and mixing significantly affect the differential distributions of Higgs production processes. For example, the transverse momentum spectrum of the Higgs in gluon fusion and $ t\bar{t}H $ production, as well as the invariant mass of the Higgs pair in double Higgs production via gluon fusion, are sensitive to the scale choice. The impact of RG running is most pronounced for the chromomagnetic dipole operator, leading to variations in the cross-section of up to $ \mathcal{O}(50\%) $. The analysis also demonstrates that the inclusion of RG running effects can significantly modify the constraints on Wilson coefficients. For instance, the fixed-scale scenario, where the system is evolved down to $ \mu = M_H $, provides the most stringent constraints. The dynamical scale scenario, where the scale is chosen based on the physical observables, and the no-running scenario, where the scale is fixed at $ \mu_0 = 1 $ TeV, yield different results. The inclusion of both linear and quadratic terms in the SMEFT analysis leads to more pronounced effects, with the allowed regions in the parameter space being significantly rotated. The results show that the combination of different Higgs production processes, such as $ H_j $, $ t\bar{t}H $, and HH, helps to substantially reduce the allowed regions in the SMEFT parameter space. The inclusion of future HL-LHC projections further improves the constraints on Wilson coefficients, particularly for the Higgs self-coupling. The study highlights the importance of considering RG running effects in the interpretation of SMEFT data, as they can significantly impact the extraction of constraints on Wilson coefficients.