We establish global rigidity upper bounds for universal determinantal point processes describing edge eigenvalues of random matrices. For this, we first obtain a general result which can be applied to general (not necessarily determinantal) point processes which have a smallest (or largest) point: this allows us to deduce global rigidity upper bounds from the exponential moments of the counting function of the process. Combining our general result with known exponential moment asymptotics for the Airy and Bessel point processes, we improve on the best known upper bounds for the global rigidity of the Airy point process, and we obtain new global rigidity results for the Bessel point process. Secondly, we obtain exponential moment asymptotics for Wright’s generalized Bessel process and the Meijer-G process, up to and including the constant term. As a direct consequence, we obtain new results for the expectation and variance of the associated counting functions. Furthermore, by combining these asymptotics with our general rigidity theorem, we obtain new global rigidity upper bounds for these point processes.