After a chain-split both chains inherit identical account states and signature rules, so a transaction signed for one is accepted by the other. Sending coins on one chain can therefore move the mirrored coins on the other without your consent.
Responsible forks add replay protection, typically a chain identifier baked into the signature so transactions cannot cross. Contentious or hasty forks sometimes do not, and users lost funds in several historical splits because of it.
If you ever hold balances across a fresh split, the safe sequence is to wait for confirmation that replay protection exists, move funds on one chain with a splitting technique if advised by the projects themselves, and treat any urgency in social media instructions as a reason for more caution rather than less.
Related: chain-split, hard-fork, double-spend, self-custody