Going by some ECB models/estimates from last year, protectionism would result in negative consequences pretty fast, e.g.
For illustration, I will use the results of simulations carried out by ECB staff using both the ECB's global model and the IMF’s multi-country model. As with all models, the uncertainties involved mean precise estimates from these scenarios should be treated with caution, but they are useful to explain the different channels at work.
To illustrate the potential effects of rising protectionism, I do not want to dwell on the specifics of the tariffs currently being discussed. This would miss the bigger picture. I rather want to consider a hypothetical scenario where the United States raises tariffs on all imports of goods by 10 percentage points, and its trading partners impose the equivalent on US exports.
According to our model simulations, such a scenario would have significant adverse effects on the global economy, including, and in particular, on the economy that raises tariffs in the first place. Specifically, real economic activity in the United States could be up to 2½% lower than in the baseline in the first year alone. [Gives some reasons for this.]
In other words, the overall scenario is clearly a net negative for the world economy as a whole. According to ECB staff simulations, world trade in goods could fall by up to 3% already in the first year after the change in tariffs and world GDP by up to 1%. Euro area GDP would also decline, but by less than in the US.
They don't make any longer term estimates for the effects of such tariffs, just mention that the EU would have been worse of by 1/5 without the economic integration since the 1950s.
Frankly it looks doubtful that US economy is showing any of that (short-term) trouble.
Are there any theoretical models of protectionism that predict short-term gain but medium or long-term loss for a country?
(Well, there's one BIS paper that advances a sort of short-term masking effect for the current US economy by advanced shipments/stockpiling:
It’s paradoxical that the United States is starting to put obstacles in the road at a time when its economy is firing on all cylinders. Even looking through the short-term boost to GDP in the latest quarter [citing data from July 2018] – ironically enough, partly owing to companies bringing forward cross-border shipments to skirt tariffs – its short-term prospects are promising. But in the long term, protectionism will bring not gain, but only pain. Not just for the United States, but for us all.
But this looks to me fairly anecdotal...)