Constructing smoothings of stable maps

Let $X$ be a smooth projective variety. Define a stable map $f:C\to X$ to be “eventually smoothable” if there is an embedding $X\hookrightarrow {\mathbb{P}}^N$ such that $(C,f)$ occurs as the limit of a $1$-parameter family of stable maps to ${\mathbb{P}}^N$ with smooth domain curves. Via an explicit deformation-theoretic construction, we produce a large class of stable maps (called “stable maps with model ghosts”), and show that they are eventually smoothable.