IPA: /ˌɪn.kənˈsɛk.wənt.li/
KK: /ɪnˈkɑnsɪkwəntli/
In a manner that does not have any significant effect or importance.
He spoke inconsequently, as if his words did not matter at all.
Inconsequently → It is formed from "in-" (meaning not) and "consequent" (from Latin "consequentem", meaning following or resulting). The word "inconsequently" means not following logically or without a logical result.
Think of 'not' ('in-') leading to something that 'follows' ('consequent') — that's why inconsequently means something that does not logically follow.