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 $X\hookrightarrow P^N$ such that $(C,f)$ occurs as the limit of a 1-parameter family of stable maps to $P^N$ 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.