The answer abx gives is completely correct; I am just adding an example proving that some hypothesis is necessary. Let $k$ be a field of characteristic different from $2$. Let $G$ be the algebraic subgroup of $\textbf{GL}_{2,k}$ of the form
$$
G =\left\{ \left[ \begin{array}{rr} 1 & b \\ 0 & \lambda \end{array} \right] \left\vert b\in \mathbf{G}_a,\ \lambda^2 = 1 \right. \right \}.
$$
Denote by $U$ the connected component of the identity, which is isomorphic to $\mathbf{G}_a$ via the coordinate $b$. This is a normal closed subgroup of $G$ whose quotient is $\mu_2 = \{1,-1\}$. Denote by $U_{-1}$ the other connected component of $G$, which is isomorphic to $\mathbf{G}_a$ via the coordinate $b_{-1}$.
Let $\Delta$ denote $\mathbb{A}^1_k$ with coordinate $t$. Let $\Delta^*$ denote the basic open affine $D(t)$ inside $\Delta$. Let $\overline{U}$ denote $\mathbb{P}^1$ with homogeneous coordinates $[b_0,b_1]$; identify $U$ with the open subset $D_+(b_0)$ via $b=b_1/b_0$.
Inside $\Delta\times_k \overline{U}$, form the blowing up of the closed point $p$ where $t=0$ and where $b_0 = 0$. Next, on the exceptional divisor $E$ with homogeneous coordinates $[t,b_0/b_1]$, form the blowing up of the point $q$ with $[t,b_0/b_1] = [1,1]$, i.e., the zero scheme of $t(b_0/b_1)^{-1} - 1$. Denote the composite of these two blowings up by
$$
\nu: \overline{X} \to \Delta \times_k \overline{U}.
$$
Denote by $\overline{F}$ the exceptional divisor over $q$ with homogeneous coordinates $[t, t(b_0/t_1)^{-1} -1]$.
Denote by $\widetilde{U}_0$ the inverse image in $\overline{X}$ of $\{0\}\times U$. Denote by $H$ the strict transform in $\overline{X}$ of the "horizontal cross-section", $\Delta\times\{[0,1]\}$. Denote by $\widetilde{E}$ the strict transform of $E$ in $\overline{X}$. Finally, denote by $F$ the open affine complement in $\overline{F}$ of the intersection point $\overline{F}\cap \widetilde{E}$; this is isomorphic to $\mathbb{A}^1_k$ via the coordinate
$$
b_F = t^{-1}(t(b_0/b_1)^{-1} -1).
$$
Denote by $X$ the open complement in $\overline{X}$ of the Cartier divisor $\widetilde{E} + H$. This is a scheme over $\Delta$, and the fiber over $t=0$ is the disjoint union $\widetilde{U} = \text{Spec}(k[b])$ and $F=\text{Spec}(k[b_F])$.
Because the blowings up occur over points of $\Delta\times\{[0,1]\}$, the open immersion of the complement extends to an open immersion,
$$
q_U: \Delta \times_k U \hookrightarrow X.
$$
Next, consider the morphism,
$$
q'_{-1}: \Delta^* \times_k U_{-1} \to \Delta \times_k \overline{U}, \ \ (t,b_{-1}) \mapsto (t,b_{-1} + t^{-1}).
$$
This extends to a unique open immersion,
$$
q_{-1}:\Delta\times_k U_{-1} \to X.
$$
In fact, $\overline{X}$ is the minimal blowing up of $\Delta\times_k \overline{U}$ such that $q'_{-1}$ extends to an open immersion.
This open immersion maps $\{0\}\times U_{-1}$ isomorphically to $F$ with identification of coordinates $b_{-1}\leftrightarrow b_F$.
The union of these two morphisms defines a morphism
$$
q: \Delta\times_k G \to X.
$$
This is a surjective, quasi-finite morphism of $\Delta$-schemes. In fact, this is a quotient of the group scheme $\Delta \times_k G$ over $\Delta$ by the closed subgroup scheme $\Gamma$ whose intersection with $\Delta\times_k U$ equals $\Delta \times_k \{0\}$ and whose intersection with $\Delta\times_k U_{-1}$ is $\{(t,b_{-1}) | tb_{-1} + 1 =0\}$.
There is an action of $G$ on $\Delta^*\times_k U$ by
$$
b*(t,\beta) = (t,b+\beta), \ \ b_{-1}*(t,\beta) = (t,-b_{-1} - t^{-1}-\beta).
$$
This extends to an action of $G$ on $X$. For $t\neq 0$, the stabilizer of $(t,0)$ is the fiber of $\Gamma$ over $t$. For $t=0$, the stabilizer in $\widetilde{U}$ of the point with $b=0$ is the identity subgroup.
In this example, the action of $G$ on $X$ is not closed. Equivalently, the morphism $q$ is not finite.
