It's listed in OED at sense 4:
4. transitive. To put or send (persons or things) away, in, off, out, etc., hurriedly and unceremoniously. Cf. ‘pack off’, ‘send packing’.
1823 W. Scott Peveril I. viii. 187 I will bundle away her rags to the Hall.
1830 T. De Quincey Life R. Bentley in Blackwood's Edinb. Mag. Sept. 438/2 When he and his are all bundled off to Hades.
1857 D. Livingstone Missionary Trav. S. Afr. xvi. 300 She..bundled him into the hut.
1876 E. Jenkins Blot on Queen's Head 5 They were bundled out pretty quick.
1878 C. Bethell Let. in Law Rep.: Weekly Notes (1887) 29 Jan. 18/1 I have been bundled off to the Cape for a year.
---and in M-W Unabr. at sense 2:
2 : to hustle or hurry unceremoniously often by shoving or throwing
- bundled the children off to school
- he bundled his possessions into an empty carriage — David Garnett
---i.e., the root verb is "to bundle" and it can be followed by various prepositions: "out," "off," "into," "away."