When you are talking about transcription, there are no amino acids(yet). In Eukaryotes, you have pre-mRNA. Part of the genomic DNA is transcribed by RNA polymerase, that have less proofreading activity than DNA polymerase, making them faster.
Next, the pre-mRNA is spliced. If a mutation occurs at the splice sites, the mRNA can contain non-coding sequences and lose coding sequences. If there is a premature stop codon, the protein will be truncated. You get half of the correct protein. It will likely not fold the way it did as part of the original protein.
If there is an insertion or deletion, there will be a frame shift, and all amino acids will be different. You get a random protein.
A mutation can also change the expression of the protein. This is regulated on several levels, including on how much protein is produced from mRNA.
If one amino acid is changed, one amino acid changes. This usually has no effect at all, as not all amino acids in a protein are important and many mutations can exchange one amino acid for a similar one. Still, single amino acid mutations can make a protein not fold properly, not able to carry out it's intended function, make the proteins aggregate, causing fibers that disrupt the cell.
A single mRNA only has a limited life time. It will be translated 0 to 100,000 times. Then the mRNA is broken down and the mutation is lost.
All protein are recycled over time, using ubiquitin, proteases and lysosomes. Protein synthesis and degradation need to be in balance. So both happen at equal rates.
The amount of mutant proteins from a somatic mutation in DNA or mRNA is usually small. Still somatic mutations can happen and can cause disease; ie cancer.
Misfolded protein diseases also exist. For those, you don't even need a mutant protein.Antibodies can recognize mutant proteins. Very early on in the embryo, all immune cells that make antibodies that bind to self protein are eliminated. So the immune system may recognize mutant protein, as it indeed does with cancer.
Usually mutant protein will just float around in the cell and be harmless. Remember that there are two copies of each allele. So there is a backup. Half the protein made is functional, the other half is mutant. If this is a germline mutation, having half the protein you normally have in all cells can cause a different phenotype or disease. Some mutations are dominant. You function fine/have the non-mutant phenotype as long as you have one good copy.