  We give several results concerning the connected component ${\rm Aut}_X^0$ of
the automorphism scheme of a proper variety $X$ over a field, such as its
behaviour with respect to birational modifications, normalization, restrictions
to closed subschemes and deformations. Then, we apply our results to study the
automorphism scheme of not necessarily Jacobian elliptic surfaces $f: X \to C$
over algebraically closed fields, generalizing work of Rudakov and Shafarevich,
while giving counterexamples to some of their statements. We bound the
dimension $h^0(X,T_X)$ of the space of global vector fields on an elliptic
surface $X$ if the generic fiber of $f$ is ordinary or if $f$ admits no
multiple fibers, and show that, without these assumptions, the number
$h^0(X,T_X)$ can be arbitrarily large for any base curve $C$ and any field of
positive characteristic. If $f$ is not isotrivial, we prove that ${\rm Aut}_X^0
\cong \mu_{p^n}$ and give a bound on $n$ in terms of the genus of $C$ and the
multiplicity of multiple fibers of $f$. As a corollary, we re-prove the
non-existence of global vector fields on K3 surfaces and calculate the
connected component of the automorphism scheme of a generic supersingular
Enriques surface in characteristic $2$. Finally, we present additional results
on horizontal and vertical group scheme actions on elliptic surfaces which can
be applied to determine ${\rm Aut}_X^0$ explicitly in many concrete cases.
