In a person with functioning innate immunity, mutated virus will be eliminated as effectively as original strain.
But the point is that innate immunity is inefficient at eliminating the virus, hence the need for acquired immunity.
Therefore if transmitted to such a person, there will be less chance that he will pass it on.
He or she - but nonetheless, innate immunity is unlikely to prevent viral replication, expulsion and therefore potential transmissibility in all but the lowest of viral load infections. But the point is that if the virus replicates in the host and is expelled (e.g. through breathing out!!) then the mutant strain is much more likely to infect a new person than the original strain. So the sequence goes as follows:
1. So for one person who becomes infectious having been exposed to both mutant and original strain they may infect four person but only one of those people becomes infected with both strains (call this person a) the other three people (call them people b, c and d) with the new strain only (as it is more transmissible).
2. Those people b, c and d - they infect four additional people all with the mutant strain, while person a infects one person with both strains and three with the mutant strain.
So from a single person infected with both strains we have 16 people infected with the mutant strain and just one infected with the original strain. That is how the mutant becomes dominant.