In this article, first of all we introduced a new concept for the $(p,q;\vartheta)$-extended Bessel-Wright function $J_{\omega;p,q}^{\sigma;\varsigma,\lambda}(z;\vartheta)$, and further discussed some new properties related to Marichev-Saigo-Maede fractional integral as well as their derivative operators, and also studied about the Caputo-type Marichev-Saigo-Maede fractional integral and its derivative operators which are applied to the $(p,q;\vartheta)$-extended Bessel-Wright function. Further, we have also discussed some special cases such as particular results on Saigo, Riemann-Liouville and Erdeyi-Kober fractional integrals and their derivative operators are obtained. As part of applications of the new $(p,q;\vartheta)$-extended Bessel-Wright function $J_{\omega;p,q}^{\sigma;\varsigma,\lambda} (z;\vartheta)$ to the fractional kinetic equations are also discussed in brief suggesting its possible solutions, please note that our current findings are based upon earlier findings of several researchers on various special functions such as Mittag-Leffler-type, $K$-type, $H$-type, $I$-type Bessel-type, Aleph-type, $S$-type, hypergeometric-type, and plenty of others (see Kiryakova \cite{AK3, AK6, AK8}), and integral transform such as Laplace, Sumudu were used by different researchers to study fractional kinetic equations.