Short Answer:
Probably. For some goods. I guess.
Longer Answer:
Assuming two toy economies and three goods, increasing the supply of good in one economy relative to the supply of goods of the other two classes will decrease its trade value relative to them.
Prior to the transfer (at $t_0$) they'd look like:
$$E_1(t_0)=\{a_1x,\ b_1y,\ c_1z\}$$
$$E_2(t_0)=\{a_2x,\ b_2y,\ c_2z\}$$
with basic prices of $x$ in $E_2$ as:
$$p_x(y)=\frac{a_2}{b_2},\ \ p_x(z)=\frac{a_2}{c_2}$$
in that it if we started with $a_1=4,$ $b_1=8,$ and $c_1=16,$ then one unit of $x$ would cost either $2y$ or $4z$ in $E_1$.
After the transfer (let's say of good $x$ of quantity $a_t$ at $t_1$, they'd look like:
$$E_1(t_1)=\{(a_1-a_t)x,\ b_1y,\ c_1z\}$$
$$E_2(t_1)=\{(a_2+a_t)x,\ b_2y,\ c_2z\}$$
consequently in the recipient country ($E_2$), the price of good $y$ in terms of $x$ would have declined to $\frac{b_2}{a_2+a_t}$. But, with a rational expectation of future gifts of good $x$, the value of $x$ in $E_2$ should also be diminished (or price of a good in terms of $x$ increased) by the NPV of the perpetuity times the probability of default at each payment, though we'll skip quantifying that for now (and merely assume it's positive and non-zero), and simply say that at $t_1$ in $E_2$ the price of $y$ in terms of $x$ is:
$$p_x(y)>\frac{a_2+a_t}{b_2}>\frac{a_2}{b_2}$$
and the price of $z$ in terms of $x$ is now:
$$p_x(z)>\frac{a_2+a_t}{c_2}>\frac{a_2}{c_2}$$
So as long as $x$ is currency, under the classical regime, any increase in the quantity of $x$ will decrease it's relative value in terms of the other commodities in the economy.
Still Longer Answer
This simplistic explanation over looks a few key factors, and can be expanded on by including basic international trade.
If $z$ is a fungible, non-perishable, and easily transported, and both economies have relatively free trade, then we have another story.
In that case, members of $E_1$ and $E_2$ would presumably have access to all the $z$ in each other's markets, so the price of $z$ in terms of $x$ would be globally set, and remain unchanged, since in the aggregate, the supply of $x$ has remained unchanged.
An Elaboration on Regulatory Dynamics
Further complicating this, however, would be any regulatory trade inefficiencies, like tariffs. Let's say $E_1$ has a tariff on the export of $z$ of $q\%$, then prices of $z$ are likely to rise in both $E_1$ and $E_2$ as the supply of $x$ has decreased in $E_1$. Since the ratios of $z:y$ should also remain unchanged if $y$ is not traded in this example, then prices of $y$ in terms of $x$ would also have to increase.
One would assume that given that their relative scarcity ($\frac{b}{c}$) hasn't changed from $t_0$ to $t_1$ (before and after the transfer), the ratio of their prices hasn't, or more formally:
$$p_{x_{t_0}}(y):p_{x_{t_0}}(z) = p_{x_{t_1}}(y):p_{x_{t_1}}(z)$$
should hold for both $E_1$ and $E_2$
Transportation and Storage Costs
Of course most consumer spending is on consumable goods (funny how they got that name), which are generally neither free to ship, undifferentiated, or truly non-perishable. In fact, in aid recipient nations, the percent of GDP that is spent on household consumption is often larger than the total GDP, precisely because of foreign aid, although speculative debt buying in developing markets has historically also been something of a problem (See Carbuagh International Economics [citation needed]).
What this means, is that there would be a price gradient sloping upward away from the border into $E_2$ and downward into the territory of $E_1$. The magnitude of the slope would of course be proportional to the ratio of $a_t$ to $a_1$ in $E_1$ and $a_2$ in $E_2$
Ricardo and Comparative Advantage
Given the relative production rates of goods $y$ and $z$ in the different economies, the prices in real currency might fluctuate away from even the trade influenced prices, as the relative supply of these goods changed inside their respective economies. Further, the country most efficient at producing one or the other of these goods would probably specialize, thereby reducing the price of that good at home in terms of $x$, and increasing the price of the other.
Inflation
Of course the Monetary Base can also increase in either economy, as represented by a growth in the quantity of $x$, further confounding the relationship of $a_t$ to the function $P_x()$. That is, assuming of course both countries are currency issuers as opposed to being part of a larger monetary bloc like the EU.
Currency Issue
Of course if our toy economies are not issuers of their own currency, or are part of some more restrictive monetary bloc, they may not have the ability to control inflation as easily.
Banking Regulation
Of course $MB$ represents only a fraction of $M_0$, which is itself only a fraction of the total Money Supply ($M$), which in our toy economies would be the respective values of $a$. Consequently, an injection of hard capital into $MB$ could have a wildly disproportionate affect on $M$. Since classic inflation is linked to $M$ through the equation of exchange, or
$$M * V = P * Q$$
Differing Local Preferences vis-a-vis Utility Functions and Environmental factors
We touched briefly on logistics earlier, but haven't really dealt with the fine-grained details of differing marginal utility. For simplicity's sake, we can use the example of ice. If we were to assume on of the goods (let's say $y$) were ice, and the recipient economy was a desert, and the donor a frozen tundra, then you could see how purchasing more ice than could be consumed between stimulus payments (in $x$) would not be beneficial.
This would mean the recipient economy would not likely see a large increase in the price of ice, unless there were adequate technology to mitigate the risk of spoilage, although that would still increase the cost of preservation, which would put downward pressure on demand.
However, if the roles were reversed, and they could just leave it sitting outside, then there would be.
Other confounding factors
And of course:
- Economies usually have more than two goods and a currency.
