The rate of sorption of n-butane on the structurally flexible metal-organic framework [Cu2(H-Me-trz-ia)2], including its complete structural transition between a narrow-pore phase and a large-pore phase, was studied by sorption gravimetry, IR spectroscopy, and powder X-ray diffraction at close to ambient temperature (283, 298, and 313 K). The uptake curves reveal complex interactions of adsorption on the outer surface of MOF particles, structural transition, of which the overall rate depends on several factors, including pressure step, temperature, as well as particle size, and the subsequent diffusion into newly opened pores. With the aid of a kinetic model based on the linear driving force (LDF) approach, both rates of diffusion and structural transition were studied independently of each other. It is shown that temperature and applied pressure steps have a strong effect on the rate of structural transition and thus, the overall velocity of gas uptake. For pressure steps close to the upper boundary of the gate-opening, the rate of structural transition is drastically reduced. This feature enables a fine-tuning of the overall velocity of sorption, which can even turn into anti-Arrhenius behavior.