Yes, you are correct.
$F_{ST}$ can vary among specific genetic regions considered. If for example there is a strong influx of females from population A to population B, then the mtDNA and the X chromosome (excluding the pseudo-autosomal region) would show much smaller $F_{ST}$ between the two populations than the Y-chromosome would.
Many other non-demographic processes can affect $F_{ST}$. Background selection (Charlesworth et al., 1993; Matthey-Doret and Whitlock, BioRKiv, 2018), selective sweep (e.g. Chen et al., 2010), heterozygote disadvantage (Ingvarsson and Whitlock, 2000), negative epistasis and other processes can affect $F_{ST}$ as well and cause $F_{ST}$ to differ among genetic regions. Even mutation rate variation (Whitlock and McCauley, 2001) can affect $F_{ST}$ (if the mutation rate for the loci considered is not ridiculously small compared to the migration rate).
For an introduction into the common misunderstandings of how $F_{ST}$ works, Whitlock and McCauley (2001) is a must read!
Just to complete this answer, would Fst computed from autosomal SNPs be fully representative of the genetic distance between two populations? Say for the influx of females from A to B, will the autosomal Fst correctly show that the admixed population is between A and B (depends on mixture proportion)?
The difficulty here is in the phrasings fully representative of the genetic distance and correctly show. There is no correct (true) genetic distance among populations, there is only the statistics of genetic distance that one wants to define and those statistics are affected by various processes.
Would an influx of female among populations affect $F_{ST}$ at autosomal loci? Yes. Would this effect be the same than at mtDNA? Not exactly, no. Is the entire parsing of genetic variation among population represented by $F_{ST}$? No, $F_{ST}$ is just a summary statistic (just like a mean or a variance is).
If there is a specific statistic (such as the effective migration rate, the effective population size or the location of loci involved in local adaptation for examples) that you are trying to infer from these $F_{ST}$ calculations, then maybe you should ask whether computing $F_{ST}$ by a specific method would allow you to make good inference of a specific statistic of interest.
