<p>The current series of two papers focuses on a one-dimensional reaction-diffusion equation with a free boundary, governed by the Stefan condition <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(h'(t)=-\mu u_x(t,h(t))\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>h</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mo>-</mo> <mi>μ</mi> <msub> <mi>u</mi> <mi>x</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>h</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, and a general nonlinearity in a time-periodic environment. We aim to determine the long-time behavior of the solutions by using the notion of propagating terraces, and show how the free boundary affects the propagation phenomena compared to the Cauchy problems qualitatively and quantitatively. In Part I of this series, we prove that the existence of a propagating terrace for the corresponding equation without the presence of free boundary, inherently implies the existence and uniqueness of a similar propagating terrace for the equation with a free boundary. Intriguingly, the coefficient <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mu \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> in the Stefan condition can significantly influence the shape of this terrace. In Part II of this series, under certain non-degeneracy assumptions on the nonlinearity, we prove that solutions starting from general front-like initial data approach the propagating terrace.</p>